math10 tg u2

10

Mathematics

Department of Education Republic of the Philippines

This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at [email protected].

We value your feedback and recommendations.

Teacher’s GuideUnit 2

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Mathematics – Grade 10 Teacher’s Guide First Edition 2015

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Development Team of the Teacher’s Guide

Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and Rosemarievic Villena-Diaz, PhD

Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida

Editor: Maxima J. Acelajado, PhD

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Introduction

This Teacher’s Guide has been prepared to provide teachers of Grade 10 Mathematics with guidelines on how to effectively use the Learner’s Material to ensure that learners will attain the expected content and performance standards.

This book consists of four units subdivided into modules which are further subdivided into lessons. Each module contains the content and performance standards and the learning competencies that must be attained and developed by the learners which they could manifest through their products and performances.

The special features of this Teacher’s Guide are: A. Learning Outcomes. Each module contains the content and

performance standards and the products and/ or performances expected from the learners as a manifestation of their understanding.

B. Planning for Assessment. The assessment map indicates the type of assessment and categorized the objectives to be assessed into knowledge, process/skills, understanding, and performance

C. Planning for Teaching-Learning. Each lesson has Learning Goals and Targets, a Pre-Assessment, Activities with answers, What to Know, What to Reflect on and Understand, What to Transfer, and Summary / Synthesis / Generalization.

D. Summative Test. After each module, answers to the summative test are provided to help the teachers evaluate how much the learners have learned.

E. Glossary of Terms. Important terms in the module are defined or clearly described.

F. References and Other Materials. This provides the teachers with the list of reference materials used, both print and digital.

We hope that this Teacher’s Guide will provide the teachers with the necessary guide and information to be able to teach the lessons in a more creative, engaging, interactive, and effective manner.

Unit 2 Module 3: Polynomial Functions ................................................................ 82

Learning Outcomes .............................................................................................. 82 Planning for Assessment ...................................................................................... 83 Planning for Teaching-Learning ........................................................................... 86 Pre-Assessment ................................................................................................... 87 Learning Goals and Targets ................................................................................. 87

Activity 1 .................................................................................................... 88 Activity 2 .................................................................................................... 89 Activity 3 .................................................................................................... 90 Activity 4 .................................................................................................... 90 Activity 5 .................................................................................................... 91 Activity 6 .................................................................................................... 91 Activity 7 .................................................................................................... 92 Activity 8 .................................................................................................... 94 Activity 9 .................................................................................................... 99 Activity 10 ................................................................................................ 100 Activity 11 ................................................................................................ 101 Activity 12 ................................................................................................ 102 Activity 13 ................................................................................................ 106 Activity 14 ................................................................................................ 107

Summary/Synthesis/Generalization ................................................................... 108 Summative Test ....................................................................................................... 109 Glossary of Terms ................................................................................................... 114 References Used in This Module ........................................................................ 115

Module 4: Circles ........................................................................................... 116

Learning Outcomes ............................................................................................ 116 Planning for Assessment .................................................................................... 117 Planning for Teaching-Learning ......................................................................... 123 Pre-Assessment ................................................................................................. 125 Learning Goals and Targets ............................................................................... 126

Lesson 1A: Chords, Arcs, and Central Angles................................................ 126 Activity 1 .................................................................................................. 127 Activity 2 .................................................................................................. 128 Activity 3 .................................................................................................. 129 Activity 4 .................................................................................................. 130 Activity 5 .................................................................................................. 131 Activity 6 .................................................................................................. 132 Activity 7 .................................................................................................. 132 Activity 8 .................................................................................................. 132 Activity 9 .................................................................................................. 133 Activity 10 ................................................................................................ 136 Activity 11 ................................................................................................ 136 Activity 12 ................................................................................................ 137 Activity 13 ................................................................................................ 138

Summary/Synthesis/Generalization ................................................................... 139

Lesson 1B: Arcs and Inscribed Angles ............................................................. 139 Activity 1 .................................................................................................. 140

Table of Contents

Curriculum Guide: Mathematics Grade 10

Activity 2 .................................................................................................. 141 Activity 3 .................................................................................................. 142 Activity 4 .................................................................................................. 143 Activity 5 .................................................................................................. 144 Activity 6 .................................................................................................. 145 Activity 7 .................................................................................................. 145 Activity 8 .................................................................................................. 146 Activity 9 .................................................................................................. 148 Activity 10 ................................................................................................ 151 Activity 11 ................................................................................................ 153 Activity 12 ................................................................................................ 154


Lesson 2A: Tangents and Secants of a Circle ................................................ 155 Activity 1 .................................................................................................. 155 Activity 2 .................................................................................................. 159 Activity 3 .................................................................................................. 160 Activity 4 .................................................................................................. 161 Activity 5 .................................................................................................. 162 Activity 6 .................................................................................................. 163 Activity 7 .................................................................................................. 164 Activity 8 .................................................................................................. 172


Lesson 2B: Tangent and Secant Segments ..................................................... 173 Activity 1 .................................................................................................. 173 Activity 2 .................................................................................................. 174 Activity 3 .................................................................................................. 174 Activity 4 .................................................................................................. 175 Activity 5 .................................................................................................. 175 Activity 6 .................................................................................................. 176 Activity 7 .................................................................................................. 176 Activity 8 .................................................................................................. 177 Activity 9 .................................................................................................. 179 Activity 10 ................................................................................................ 180

Summary/Synthesis/Generalization ................................................................... 180 Summative Test ....................................................................................................... 181 Glossary of Terms ................................................................................................... 189 List of Theorems and Postulates on Circles .................................................... 191 References and Website Links Used in This Module .................................... 193

Module 5: Plane Coordinate Geometry .................................................. 198

Learning Outcomes ............................................................................................ 198 Planning for Assessment .................................................................................... 199 Planning for Teaching-Learning ......................................................................... 205 Pre-Assessment ................................................................................................. 207 Learning Goals and Targets ............................................................................... 207

Lesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate Proof .................................................................... 207

Activity 1 .................................................................................................. 208 Activity 2 .................................................................................................. 208 Activity 3 .................................................................................................. 209 Activity 4 .................................................................................................. 210

Activity 5 .................................................................................................. 212 Activity 6 .................................................................................................. 212 Activity 7 .................................................................................................. 213 Activity 8 .................................................................................................. 215 Activity 9 .................................................................................................. 216 Activity 10 ................................................................................................ 217 Activity 11 ................................................................................................ 220


Lesson 2: The Equation of a Circle .................................................................... 221 Activity 1 .................................................................................................. 221 Activity 2 .................................................................................................. 222 Activity 3 .................................................................................................. 223 Activity 4 .................................................................................................. 225 Activity 5 .................................................................................................. 226 Activity 6 .................................................................................................. 227 Activity 7 .................................................................................................. 227 Activity 8 .................................................................................................. 228 Activity 9 .................................................................................................. 228 Activity 10 ................................................................................................ 229

Summary/Synthesis/Generalization ................................................................... 230 Summative Test ....................................................................................................... 231 Glossary of Terms ................................................................................................... 237 References and Website Links Used in This Module .................................... 238

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Module 3: Polynomial Functions

A. Learning Outcomes Content Standard:

The learner demonstrates understanding of key concepts of

polynomial functions. Performance Standard:

The learner is able to conduct systematically in different fields

a mathematical investigation involving polynomial functions.

Unpacking the Standards for Understanding

Subject: Mathematics 10

Quarter: Second Quarter

TOPIC: Polynomial Functions

Lesson: Illustrating Polynomial Functions, Graphs of Polynomial Functions and Solutions of Problems Involving Polynomial Functions

Learning Competencies

1. Illustrate polynomial functions 2. Graph polynomial functions 3. Solve problems involving

polynomial functions

Writer:

Elino Sangalang Garcia

Essential Understanding: Students will understand that polynomial functions are useful tools in solving real-life problems and in making decisions given certain constraints.

Essential Question: How do the mathematical concepts help solve real-life problems that can be represented as polynomial functions?

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Transfer Goal:

Students will be able to apply the key concepts of polynomial functions in finding solutions and making decisions for certain life problems.

B. Planning for Assessment

Product/Performance

The following are products and performances that students are expected to come up with in this module. 1. Write polynomial functions in standard form 2. List all intercepts of polynomial functions written in both standard and

factored forms 3. Make a list of ordered pairs of points that satisfy a polynomial function 4. Make a table of signs for polynomial functions 5. Make a summary table of properties of the graph of polynomial functions

(behavior, number of turning points, location relative to the x-axis) 6. Formulate and solve real-life problems applying polynomial functions 7. Sketch plans or designs of objects that illustrate polynomial functions g. Create concrete objects as products of applying solutions to problems

involving polynomial functions (e.g. rectangular open box, candle mold)

Assessment Map

TYPE KNOWLEDGE PROCESS/ SKILLS UNDERSTANDING PERFORMANCE

Pre-Assessment/ Diagnostic

Part I Illustrating polynomial functions (Recalling the definition of polynomial functions and the terms associated with it)

Part I Illustrating polynomial functions (Recalling the definition of polynomial functions and the terms associated with it) Graphing polynomial functions (Describing the properties of graphs of polynomial functions)

Part I Graphing polynomial functions (Describing the properties of graphs of polynomial functions) Solving problems involving polynomial functions

Part II Products and performances related to or involving quadratic functions (Solving area problems)

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Formative Quiz 1: Illustrating polynomial functions (Writing polynomial functions in standard form and in factored form)

Quiz 2: Graphing polynomial functions (Finding the intercepts of polynomial functions) (Finding additional points on the graph of a polynomial function)

Quiz 3: Graphing polynomial functions (Preparing table of signs) (Describing the behavior of the graph using the Leading Coefficient Test)

Quiz 4: Graphing polynomial functions (Identifying the number of turning points and the behavior of the graph based on multiplicity of zeros) (Sketching the graph of polynomial functions using all properties)

Quiz 5: Graphing polynomial functions (Sketching the graph of polynomial functions using all properties) Solving problems involving polynomial functions

Quiz 6: Solving problems involving polynomial functions (Solving real-life problems that apply polynomial functions)

Summative Assessment

Part I Illustrating polynomial functions (Recalling the definition of polynomial functions and the terms associated with it)

Part I Illustrating polynomial functions (Recalling the definition of polynomial functions and the terms associated with it) Graphing polynomial functions

Part I Graphing polynomial functions (Describing the properties of the graph of polynomial functions) Solving problems involving polynomial functions

Part II Products and performances related to or involving polynomial functions (Solving problems related to volume of an open rectangular box)

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(Describing the properties of the graphs of polynomial functions)

Self-Assessment

(optional)

Journal Writing: Expressing understanding of polynomial functions, graphing

polynomial functions, and solving problems involving polynomial functions

Assessment Matrix (Summative Test)

Levels of

Assessment What will I assess? How will I assess? How Will I Score?

Knowledge 15%

The learner demonstrates understanding of key

concepts of

polynomial functions. Illustrate polynomial functions. Graph polynomial functions Solve problems involving polynomial functions

Paper and Pencil Test Part I items 1, 2, and 3

1 point for every correct response

Process/Skills 25%

Part I items 4, 5, 6, 7, and 8


Understanding 30%

Part I items 9, 10, 11, 12, 13, and 14


Product/ Performance

30%

The learner is able to

conduct systematically

a mathematical investigation involving

polynomial functions

in different fields. Solve problems involving polynomial functions.

Part II (6 points)

Rubric for the Solution to the Problem Criteria: Use of polynomial

function as model Use of appropriate

mathematical concept

Correctness of the final answer

Rubric for the Output (Open Box) Criteria: Accuracy of

measurement (Dimensions)

Durability and Attributes

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C. Planning for Teaching-Learning

Introduction

This module is a one-lesson module. It covers key concepts of polynomial functions. It is composed of fourteen (14) activities, three (3) of which are for illustration of polynomial functions, nine (9) are for graphing polynomial functions, and two (2) are for solving real-life problems involving polynomial functions.

The lesson as incorporated in the activities is designed for the

students to: 1. define polynomial functions and the terms associated with it;

2. write polynomial functions in standard and factored form;

3. write polynomial functions in standard form given real numbers as coefficients and exponents;

4. recall and apply the different theorems in factoring polynomials to determine the x-intercepts;

5. determine more ordered pairs that satisfy a polynomial function;

6. investigate and analyze the properties of the graphs of polynomial functions (like end behaviors, behaviors relative to the x-axis, number of turning points, etc.); and

7. solve real-life problems (like area and volume, deforestation, revenue-advertising expense situations, etc.) that apply polynomial functions.

One of the essential targets of this module is for the students

to manually sketch the graph of polynomial functions which later on can be verified and validated with some graphing utilities like Grapes, GeoGebra, or even Geometer’s Sketchpad.

In dealing with each activity of this lesson, the students are

given the opportunity to use their prior knowledge and required skills in previous tasks. They are also given varied activities to process the knowledge and skills learned and further deepen and transfer their understanding of the different lessons.

Lastly, you may prepare your own related activities if you feel

that the activities suggested here are not appropriate to the level and contexts of students (for examples, slow/fast learners, and localized situations/examples).

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As an introduction to the main lesson, show the students the picture mosaic below, then ask them the question that follows:

In this mosaic picture, can you see some mathematical representations? Give some.

Motivate the students to find out the answers and to determine

the essential applications of polynomial functions through this module.

Objectives:

After the learners have gone through this module, they are expected to:

1. illustrate polynomial functions; 2. graph polynomial functions; and 3. solve problems involving polynomial functions.

PRE-ASSESSMENT:

Check students’ prior knowledge, skills, and understanding of mathematics concepts related to polynomial functions. Assessing these will facilitate your teaching and the students’ understanding of the lessons in this module.

LEARNING GOALS AND TARGETS:

Students are expected to demonstrate understanding of key concepts of polynomial functions, formulate real-life problems involving these concepts, and solve these using a variety of strategies. They are also expected to investigate mathematical relationships in various situations involving polynomial functions.

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What to KNOW

The students need first to recall the concept of polynomial expressions. These will lead them to define and illustrate mathematically the polynomial functions. Activity 1: Which is which? Answer Key

1. polynomial 2. not polynomial because the variable of one term is inside the radical

sign 3. polynomial 4. not polynomial because the exponents of the variable are not whole

numbers 5. not polynomial because the variables are in the denominator 6. polynomial 7. not polynomial because the exponent of one variable is not a whole

number 8. polynomial 9. not polynomial because the exponent of one variable is negative

10. polynomial

Answer Key

Part I: Part II. (Use the rubric to rate students’ work/output) Solution to the problem Since wlP 22 , then wl 2236 or wl 18 , and

lw 18 . The lot area can be expressed as )18()( lllA or

218)( lllA . )18()( 2 lllA 81)8118()( 2 lllA 81)9()( 2 llA , in vertex form. Therefore, 9l meters and 991818 lw meters, yielding the maximum area of 81 square meters.

1. B 8. B 2. C 9. A 3. A 10. A 4. D 11. D 5. A 12. D 6. D 13. A 7. C 14. A

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Let this activity be the starting point of defining a polynomial function as follows:

Other notations:

Activity 2: Fix and Move Them, Then Fill Me Up Answer Key

Polynomial Function Polynomial Function in Standard Form Degree Leading

Coefficient Constant

Term

1. 22112)( xxxf 2112)( 2 xxxf 2 2 2

2. xxxf 1535

32)(

3

3515

32)(

3

xxxf 3 32

35

3. )5( 2 xxy xxy 53 3 1 0

4. )3)(3( xxxy xxy 93 3 -1 0

5. 2)1)(1)(4( xxxy 4353 234 xxxxy 4 1 4

012

21

1 ...)( axaxaxaxaxf nn

nn

nn

or 01

22

11 ... axaxaxaxay n

nn

nn

n

,

A polynomial function is a function of the form

012

21

1 ...)( axaxaxaxaxP nn

nn

nn

, ,0na

where n is a nonnegative integer, naaa ...,,, 10 are real numbers called coefficients, n

nxa is the leading term, na is the leading coefficient, and 0a is the constant term.

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Activity 3: Be a Polynomial Function Architect

The answers above are expected to be given by the students. In addition, instruct them to classify each polynomial according to the degree. Also, let them identify the leading coefficient and the constant term. What to PROCESS

In this section, the students need to revisit the lessons and their knowledge on evaluating polynomials, factoring polynomials, solving polynomial equations, and graphing by point-plotting. Activity 4: Do you miss me? Here I Am Again

The preceding task is very important for the students because it has something to do with the x-intercepts of a graph. These are the x-values when y = 0, and, thus the point(s) where the graph intersects the x-axis can be determined.

Answer Key

1. )2(3)1( xxx 6. )4)(3( xxxy

2. )3)(3(23 xxxx 7. )4)(2)(2( 2 xxxy

3. (2x -3) x -1 (x -3) 8. )3)(1)(1)(1(2 xxxxy

4. )3)(2)(2( xxx 9. )3)(3)(1)(1( xxxxxy 5. )3)(2)(1)(32( xxxx 10. )3)(2)(1)(32( xxxxy

Answer Key

1. xxxxf61

472)( 23 4. xxxxf 2

61

47)( 23

2. xxxxf47

612)( 23

5. xxxxf 2

47

61)( 23

3. xxxxf612

47)( 23 6. xxxxf

472

61)( 23

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Activity 5: Seize Me and Intercept Me

Activity 6: Give Me More Companions

Answer Key

1. x-intercepts: -4, -2, 1, 3 y-intercept: 24

x -5 -3 0 2 4 y 144 -24 24 -24 144

ordered pairs: (-5,144), (-4,0), (-3, -24), (-2,0), (0,24), (1,0),

(2-24), (3,0), (4,144)

2. x-intercepts: -5, 23

, 2, 4

y-intercept: -90

x -6 -4 -0.5 3 5 y -720 240 -101.2 72 -390

ordered pairs: (-6, -720), (-5, 0), (-4, 240), (23

, 0), (-0.5, 101.2),

(2, 0), (3, 72), (4, 0), (5, -390)

3. x-intercepts: -6, 0, 34

y-intercept: 0

x -7 -3 1 2 y 175 -117 7 -32

ordered pairs: (-7,175), (-6,0), (-3,-117), (0,0), (1,7), (34

,0),

Answer Key

1. x-intercepts: 0, -4, 3 2. x-intercepts: 2, 1, -3 3. x-intercepts: 1, -1, -3 4. x-intercepts: 2, -2 5. x-intercepts: 0, 1, -1, -3, 3

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(2,-32)

4. x-intercepts: -3, -1, 0, 1, 3 y-intercept: 0

x -4 -2 -0.5 0.5 2 4 y 1680 -60 1.64 1.64 -60 1680

ordered pairs: (-4,1680), (-3, 0), (-2, -60), (-1, 0), (-0.5, 1.64),

(0, 0), (0.5, 1.64), (1, 0), (2, -60), (3, 0), (4, 1680)

Activity 7: What is the destiny of my behavior?

Answer Key

Value of x

Value of y

Relation of y-value to 0:

0, 0, or 0y y y ?

Location of the Point (x,y): above the x-

axis, on the x-axis, or below the x-axis?

-5 144 0y above the x-axis -4 0 y = 0 on the x - axis -3 -24 0y below the x-axis -2 0 y = 0 on the x - axis 0 24 0y above the x-axis 1 0 y = 0 on the x - axis 2 -24 0y below the x-axis 3 0 y = 0 on the x - axis 4 144 0y above the x-axis

Answers to the Questions:

1. (-4,0), (-2,0), (1,0), and (3,0) 2. The graph is above the x-axis. 3. The graph is below the x-axis. 4. The graph is above the x-axis. 5. The graph is below the x-axis. 6. The graph is above the x-axis.

Show the students how to prepare a simpler but similar table, the table of signs.

93

Example:

The roots of the polynomial function )3)(1)(2)(4( xxxxy are x = -4, -2, 1, and 3 . These are the only values of x where the graph will cross the x-axis. These roots partition the number line into intervals. Test values are then chosen from within each interval.

Intervals 4x 24 x 12 x 31 x 3x

Test Value -5 -3 0 2 4 4x – + + + + 2x – – + + + 1x – – – + + 3x – – – – +

)3)(1)(2)(4( xxxxy + – + – + position of the curve relative to the x-axis above below above below above

Give emphasis that at this level, though, we cannot yet determine

the turning points of the graph. We can only be certain that the graph is correct with respect to intervals where the graph is above, below, or on the x-axis as shown on the next page.

94

Activity 8: Sign on and Sketch Me Answer Key

1. )4)(1)(32( xxxy

(a) 23

, 1, 4

(b) 23

x , 123

x , 41 x , 4x

(c)

Intervals

23

x 123

x 41 x 4x

Test Value -2 0 2 5 32 x - + + +

1x - - + + 4x - - - +

)4)(1)(32( xxxy – + – + position of the curve relative to the x-axis below above below above

(d)

2. 12112 23 xxxy or )4)(1)(3( xxxy

(a) -3, 1, 4

(b) 3x , 13 x , 41 x , 4x

95

(c)

Intervals 3x 13 x 41 x 4x

Test Value -4 0 2 5 3x - + + + 1x - - + + 4x - - - +

)4)(1)(3( xxxy + - + - position of the curve relative to the x-axis above below above below

Note: Observe that there is one more factor, -1, that affects the final sign of y. For example, under

3x , the sign of y is positive because = +-(-)(-)(-) .

(d)

3. 2526 24 xxy or )5)(1)(1)(5( xxxxy

(a) -5, -1, 1, 5

(b) 5x , 15 x , 11 x , 51 x , 5x

(c)


Test Value -6 -2 0 2 6 5x - + + + + 1x - - + + + 1x - - - + + 5x - - - - +

2526 24 xxy + – + – + position of the curve relative to the x-axis

above below above below above

96

(d)

4. 101335 234 xxxxy or

2)1)(2)(5( xxxy

(a) -5, -2, 1

(b) 5x , 25 x , 12 x , 1x

(c)

Intervals 5x 25 x 12 x 1x

Test Value -6 -3 0 2 5x - + + + 2x - - + +

2)1( x + + + + 2)1)(2)(5( xxxy - + - -

position of the curve relative to the x-axis below above below below

Note: Observe that there is one more factor, -1, that affects the final sign of y. For example, under 5x , the sign of y is negative because = --(-)(-)(+) . .

(d)

97

5. 342 )1()1)(3( xxxxy

(a) -3, -1, 0, 1

(b) 3x , 13 x , 01 x , 10 x , 1x

(c)


Test Value -4 -2 -0.5 0.5 2 2x + + + + + 3x - + + + +

4)1( x + + + + + 3)1( x - - - - +

342 )1()1)(3( xxxxy + – – – + position of the curve relative to the x-axis above below below below above

(d)

Broken parts of the graph indicate that somewhere below, they are connected. The graph goes downward from (-1,0) and at a certain point, it turns upward to (-3,0).


1. For )4)(1)(32( xxxy

a. Since there is no other x-intercept to the left of 23

, then the

graph falls to the left continuously without end.

b. (i) 123

x and 4x (ii) 23

x and 41 x

c. Since there is no other x-intercept to the right of 4, then the graph rises to the right continuously without end.

d. leading term: 32x e. leading coefficient: 2, degree: 3

98

2. For 12112 23 xxxy or )4)(1)(3( xxxy a. Since there is no other x-intercept to the left of -3, then the

graph rises to the left continuously without end. b. (i) 3x and 41 x (ii) 13 x and 4x c. Since there is no other x-intercept to the right of 4, then the

graph falls to the right continuously without end. d. leading term: 3x e. leading coefficient: -1, degree: 3

3. For 2526 24 xxy or )5)(1)(1)(5( xxxxy

a. Since there is no other x-intercept to the left of -5, then the graph rises to the left continuously without end.

b. (i) 5x and 11 x (ii) 15 x and 51 x c. Since there is no other x-intercept to the right of 5, then the

graph rises to the right continuously without end. d. leading term: 4x e. leading coefficient: 1, degree: 4

4. For 101335 234 xxxxy or 2)1)(2)(5( xxxy

a. Since there is no other x-intercept to the left of -5, then the graph falls to the left continuously without end.

b. (i) 25 x (ii) 5x , 12 x and 1x c. Since there is no other x-intercept to the right of 1, then the

graph falls to the right continuously without end. d. leading term: 4x e. leading coefficient: -1, degree: 4

5. For 342 )1()1)(3( xxxxy

a. Since there is no other x-intercept to the left of -3, then the graph rises to the left continuously without end.

b. (i) 3x and 1x (ii) 13 x , 1 0,x and 10 x c. Since there is no other x-intercept to the right of 1, then the

graph rises to the right continuously without end. d. leading term: 10x e. leading coefficient: 1, degree: 10

Let the students reflect on these questions: Do the leading

coefficient and degree of the polynomial affect the behavior of its graph? Encourage them to do an investigation as they perform the next activity.

99

Activity 9: Follow My Path! Answer Key Case 1: a. positive b. odd degree c. falling to the left rising to the right Case 2: a. negative b. odd degree c. rising to the left falling to the right Case 3: a. positive b. even degree c. rising to the left rising to the right Case 4: a. negative b. even degree c. falling to the left falling to the right Summary table:

Sample Polynomial Function

Leading Coefficient:

0n or

0n

Degree: Even

or Odd

Behavior of the Graph: Rising or

Falling Possible Sketch

Left-hand

Right-hand

1. 12772 23 xxxy

0n odd falling rising

2. 473 2345 xxxxy

0n odd rising falling

3. xxxy 67 24 0n even rising rising

4. 2414132 234 xxxxy

0n even falling falling

100

Synthesis: (The Leading Coefficient Test)

1. If the degree of the polynomial is odd and the leading coefficient is positive, then the graph falls to the left and rises to the right.

2. If the degree of the polynomial is odd and the leading coefficient is negative, then the graph rises to the left and falls to the right.

3. If the degree of the polynomial is even and the leading coefficient is positive, then the graph rises to the right and also rises to the left.

4. If the degree of the polynomial is even and the leading coefficient is negative, then the graph falls to the left and also falls to the right.

You should also consider another helpful strategy to determine whether the graph crosses or is tangent to the x-axis at each x-intercept. This strategy involves the concept of multiplicity of a root of a polynomial function, the one generalized in the next activity. Activity 10: How should I pass through? Answer Key

Root or Zero Multiplicity

Characteristic of

Multiplicity: Odd or even

Behavior of Graph Relative to x-axis at this Root:

Crosses or is Tangent to

-2 2 even tangent to x-axis -1 3 odd crosses the x-axis 1 4 even tangent to x-axis 2 1 odd crosses the x-axis

Answer to the Questions:

a. The graph is tangent to the x-axis. b. The graph crosses the x-axis.

The next activity considers the number of turning points of the graph of a polynomial function. The turning points of a graph occur when the function changes from decreasing to increasing or from increasing to decreasing values.

101

Activity 11: Count Me In Answer Key

Polynomial Function Sketch Degree

Number of

Turning Points

1. 4xy

4 1

2. 152 24 xxy

4 3

3. 5xy

5 0

4. 1235 xxxy

5 2

x

y

x

y

x

x

102

5. xxxy 45 35

5 4


a. Quartic functions: have an odd number of turning points; at most 3 turning points Quintic functions: have an even number of turning points; at most 4 turning points

b. No. It is not possible. c. The number of turning points is at most (n – 1).

Important: The graph of a polynomial function is continuous, smooth, and

has rounded turns. What to REFLECT on and UNDERSTAND Activity 12: It’s Your Turn, Show Me Answer Key

1. )52()1)(3( 2 xxxy a. leading term: 42x b. end behaviors: rises to the left, falls to the right

c. x-intercepts: -3, -1, 25

points on x-axis: (-3,0), (-1,0), (25

,0)

d. multiplicity of roots: -3 has multiplicity 1, -1 has multiplicity 2,

25

has multiplicity 1

e. y-intercept: 15 point on y-axis: (0,15) f. no. of turning points: 1 or 3

y

x

y

103

g. expected graph:

Note: At this stage, we cannot determine the exact values of all the turning points of the graph. We need calculus for this. For now, we just need to ensure that the graph's end behaviors and intercepts are correctly graphed.

2. 322 )2()1)(5( xxxy

a. leading term: 7x b. end behaviors: falls to the left, rises to the right c. x-intercepts: 5 , 1, 5 , 2 points on the x-axis: ( 5 ,0), (1,0), ( 5 ,0), (2,0) d. multiplicity of roots: 5 has multiplicity 1, 1 has multiplicity 2, 5 has multiplicity 1, 2 has multiplicity 3 e. y-intercept: 40 point on y-axis: (0, 40) f. no. of turning points: 2 or 4 or 6 g. expected graph:

Note: Broken parts of the graph indicate that somewhere above, they are connected. The graph goes upward from (1, 0) and at a certain point, it turns downward to ( 5 , 0).

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3. 422 23 xxxy or in factored form )2)(2( 2 xxy a. leading term: 3x b. end behaviors: rises to the left, falls to the right c. x-intercept: 2 point on x-axis: (2, 0) d. multiplicity of root: -2 has multiplicity 1 e. y-intercept: 4 point on y-axis: (0, 4) f. no. of turning points: 0 or 2 g. expected graph:

Note: The graph seems to be flat near x = 1. However, at this stage, we cannot determine whether there are any “flat” parts in the graph. We need calculus for this. For now, we just need to ensure that the graph's end behaviors and intercepts are correctly graphed.

4. )32)(7( 22 xxxy a. leading term: 52x b. end behaviors: falls to the left, rises to the right

c. x-intercepts: 7 , 23

, 0, 7

points on the x-axis: ( 7 , 0), (23

, 0), (0, 0), ( 7 , 0)

d. multiplicity of roots: 7 has multiplicity 1, 23

has

multiplicity 1, 0 has multiplicity 2, 7 has multiplicity 1

e. y-intercept: 0 point on the y-axis: (0, 0) f. no. of turning points: 2 or 4

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g. expected graph:

5. 2861832 234 xxxxy or in factored form )2)(72)(2( 2 xxxy

a. leading term: 42x b. end behaviors: rises to the left, rises to the right

c. x-intercepts: -2, 2 , 2 , 27

points on x-axis: (-2, 0), ( 2 , 0), ( 2 , 0), (27

, 0)

d. multiplicity of roots: -2 has multiplicity 1, 2 has multiplicity 1,

2 has multiplicity 1, 27

has multiplicity

1 e. y-intercept: 28 point on y-axis: (0, 28) f. no. of turning points: 1 or 3 g. expected graph:

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Activity 13: Investigate Deeper and Decide Wisely

Answers to the Questions

1. a. 50% b. The value given by the table is 23.7%. The polynomial gives a

value of 26.3%. The given polynomial is the cubic polynomial that best fits the data. We expect it to give a good approximation of the forest cover but it may not necessarily produce the exact values.

c. The domain of the function is [0,98]. Since year 2100 corresponds to x = 200, we cannot use the function to predict forest cover during this year. Moreover, if x = 200, the polynomial predicts a forest cover of 59.46%. This is very unrealistic unless major actions are done to reverse the trend.

You can find other data that can be modelled by a

polynomial. Use the regression tool in MS Excel or GeoGebra to determine the best fit polynomial for the data.

2. The figure below can help solve the problem.

18 - 2x18

24 - 2x

24

x

x

x

x

x

x

x

x

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Solution: Let x be the height of the box 18 – 2x be the width of the box 24 – 2x be the length of the box Working Equation: Vlwh

)()218)(224( xVxxx 560)218)(224( xxx 560432844 23 xxx

0560432844 23 xxx 014010821 23 xxx 0)14)(5)(2( xxx

To meet the requirements, the height of the box is either 2 inches or 5 inches. Both will result in the volume of 560 cubic inches. In this problem, it is impossible to produce a box if the height is 14 inches, so x = 14 is not a solution.

Encourage the students to write their insights. Let them show their appreciation of polynomial functions. The following questions might be helpful for them: Were you surprised that polynomial functions have real and practical uses? What mathematical concepts do you need to solve these kinds of problems? What to TRANSFER The goal of this section is to check if the students can apply polynomial functions to real-life problems and produce a concrete object that satisfies the conditions given in the problem. Activity 14: Make Me Useful, Then Produce Something Answers to the Questions Solution: Let x be the side of the square base of the pyramid. So, area of the base (B): 2xB height of the pyramid (h): 2 xh

Working Equation: BhV31

108

)2(31)( 2 xxxV

)2(3125 2 xx

23 275 xx 0752 23 xx 0)153)(5( 2 xxx The only real solution to the equation is 5. So, the side of

the square base is 5 inches long and the height of the pyramid is 3 inches.

Students’ outputs may vary depending on the materials used and in the way they consider the criteria. Summary/Synthesis/Generalization: This lesson was about polynomial functions. You learned how to:

illustrate and describe polynomial functions;

show the graph of polynomial functions using the following properties: - the intercepts (x-intercept and y-intercept); - the behavior of the graph using the Leading Coefficient Test,

table of signs, turning points, and multiplicity of zeros; and

solve real-life problems that can be modelled with polynomial functions.

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SUMMATIVE TEST

Part I Choose the letter that best answers each question. 1. Which of the following could be the value of n in the equation

f(x) = xn if f is a polynomial function?

A. – 2 C. 41

B. 0 D. 3

2. Which of the following is NOT a polynomial function? A. )(xf C. 35)( xxxf

B. 132)( 3 xxf D. 25

1

2)( xxxf

3. What is the leading coefficient of the polynomial function 42)( 3 xxxf ?

A. – 4 C. 1 B. – 2 D. 3

4. How should the polynomial function 342 21121)( xxxxxf be

written in standard form? A. 234

21211)( xxxxxf

B. 432 11221)( xxxxxf

C. xxxxxf21211)( 234

D. 432 11221)( xxxxxf

5. Which polynomial function in factored form represents the given

graph?

A. 2)1)(32( xxy B. 2)1)(32( xxy C. )1()32( 2 xxy D. )1()32( 2 xxy

y

110

6. Which of the following could be the graph of 45 24 xxy ?

A. C.

B. D. 7. If you will draw the graph of )1(2 xxy , how will the graph behave

at the x-axis? A. The graph crosses both (0, 0) and (1, 0). B. The graph crosses (0, 0) and is tangent to the x-axis at (1, 0). C. The graph crosses (1, 0) and is tangent to the x-axis at (0, 0). D. The graph is tangent to the x-axis at both (0, 0) and (1, 0).

8. You are asked to graph xxxxxxxf 23456 35)( using its

properties. Which of these will be your graph?

A.

-6 -5 -4 -3 -2 -1 1 x

-5

-4

-3

-2

-1

1

2

y

O

B.

-6 -5 -4 -3 -2 -1 1 x

-5

-4

-3

-2

-1

1

2

y

O

C.

-6 -5 -4 -3 -2 -1 1 x

-6

-5

-4

-3

-2

-1

1

y

O

D.

-6 -5 -4 -3 -2 -1 1 x

-6

-5

-4

-3

-2

-1

1

y

O

9. Given that 237)( xxxf n , what value should be assigned to n to

make f a function of degree 7?

A. 37

B. 73

C. 73

D. 37

y y

y y

x

x

x

x

111

10. If you were to choose from 2, 3, and 4, which pair of values for a and n would you consider so that y = axn could define the graph below?

-8 -7 -6 -5 -4 -3 -2 -1 x

-6

-5

-4

-3

-2

-1

1

2

y

O

11. A car manufacturer determines that its profit, P, in thousands of pesos, can be modeled by the function P(x) = 0.001 25x4 + x – 3, where x represents the number of cars sold. What is the profit at x =150?

A. Php 75.28 C. Php 3,000,000.00 B. Php 632,959.50 D. Php 10,125,297.00

12. Your friend Aaron Marielle asks your help in drawing a rough sketch of the graph of )32)(1( 42 xxy by means of the Leading Coefficient Test. How will you explain the behavior of the graph?

A. The graph is falling to the left and rising to the right. B. The graph is rising to both left and right. C. The graph is rising to the left and falling to the right. D. The graph is falling to both left and right.

13. Lein Andrei is tasked to choose from the numbers –2, –1, 3, and 6 to

form a polynomial function in the form y = axn. What values should he assign to a and n so that the function could define the graph below?

14. Consider this Revenue-Advertising Expense situation.

A drugstore that sells a certain brand of vitamin capsule estimates that the profit P (in pesos) is given by

320,2000240050 23 xxxP

A. a = 3 , n = -2 B. a = 3 , n = 6 C. a = 6 , n = 3 D. a = -1 , n = 6

A. a = 2 , n = 3 B. a = 3 , n = 2 C. a = 2 , n = 4 D. a = 3 , n = 3

x

y

112

where x is the amount spent on advertising (in thousands of pesos). An advertising agency provides four (4) different advertising

packages with costs listed below. Which of these packages will yield the highest revenue for the company?

A. Package A: Php 8,000.00 B. Package B: Php 16,000.00 C. Package C: Php 32,000.00 D. Package D: Php 48,000.00

Part 2 Read and analyze the situation below. Then, answer the questions or perform the required task.

An open box with dimensions 2 inches by 3 inches by 4 inches

needs to be increased in size to hold five times as much material as the current box. (Assume each dimension is increased by the same amount.)

Task:

(a) Write a function that represents the volume V of the new box. (b) Find the dimensions of the new box. (c) Using hard paperboard, make the two boxes - one with the

original dimensions and another with the new dimensions. (d) On one face of the bigger box, write your mathematical

solution in getting the new dimensions.

Additional guidelines:

1. The boxes should look presentable and are durable enough to hold any dry material such as sand, rice grains, etc.

2. Consider the rubric below.

Rubric for Rating the Output:

Point Descriptor

3 Polynomial function is correctly presented as model, appropriate mathematical concepts are used in the solution, and the correct final answer is obtained.

2 Polynomial function is correctly presented as model, appropriate mathematical concepts are partially used in the solution, and the correct final answer is obtained.

1 Polynomial function is not correctly presented as model, other alternative mathematical concepts are used in the solution, and the final answer is incorrect.

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Criteria for Rating the Output (Box):

Each box has the needed dimensions. The boxes are durable and presentable.

Point/s to be Given:

3 points if the boxes have met the two criteria 2 points if the boxes have met only one criterion

1 point if the boxes have not met any of the criteria

Answer Key for Summative Test

Part I: Part II.

(Use the rubric to rate students’ work/output) Solution for finding the dimensions of the desired box:

Let x be the number to be added to each of length, width and height to increase the size of the box. Then the dimensions of the new box are x+2 by x+3 by x+4. Since the volume of the original box is (2 inches) (3 inches) (4 inches) = 24 cubic inches, then the volume of the new box is 120 cubic inches. Writing these in an equation, we have )()4)(3)(2( xVxxx 12024269 23 xxx 096269 23 xxx , 0)4811)(2( 2 xxx Therefore, from the last equation, the only real solution is x = 2. Thus, the dimensions of the new box are 4 inches by 5 inches by 6 inches. Note to the Teacher: To validate that the volume of the bigger box is five times the volume of the other box, guide the students to compare the content of both boxes using sand, rice grains, or mongo seeds.

1. B 2. D 3. B 4. C 5. B 6. A 7. C 8. C

9. A 10. B

11. B 12. D 13. D 14. C

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Glossary of Terms

Constant Function – a polynomial function whose degree is 0 Cubic Function – a polynomial function whose degree is 3 Evaluating a Polynomial – the process of finding the value of the polynomial at a given value in its domain Intercepts of a Graph – the points on the graph that have zero as either the x-coordinate or the y-coordinate Irreducible Factor - a factor that can no longer be factored using coefficients that are real numbers Leading Coefficient Test - a test that uses the leading term of the polynomial function to determine the right-hand and the left-hand behaviors of the graph Linear Function - a polynomial function whose degree is 1 Multiplicity of a Root - tells how many times a particular number is a root for a given polynomial Nonnegative Integer - zero or any positive integer Polynomial Function - a function denoted by

012

21

1 ...)( axaxaxaxaxP nn

nn

nn

, where n is a nonnegative

integer, naaa ...,,, 10 are real numbers called coefficients, but ,0na , n

nxa is the leading term, na is the leading coefficient, and 0a is the constant term Polynomial in Standard Form - any polynomial whose terms are arranged in decreasing powers of x Quadratic Function - a polynomial function whose degree is 2 Quartic Function - a polynomial function whose degree is 4 Quintic Function - a polynomial function whose degree is 5 Turning Point - point where the function changes from decreasing to increasing or from increasing to decreasing values

115

References

Alferez, M. S., Duro, MC.A., & Tupaz, KK. L. (2008). MSA Advanced Algebra. Quezon City, Philippines: MSA Publishing House Berry, J., Graham, T., Sharp, J., & Berry, E. (2003). Schaum’s A-Z Mathematics. London, United Kingdom: Hodder &Stoughton Educational. Cabral, E. A., De Lara-Tuprio, E. P., De Las Penas, ML. N., Francisco, F. F., Garces, IJ. L., Marcelo, R. M., & Sarmiento, J. F. (2010). Precalculus. Quezon City, Philippines: Ateneo de Manila University Press Jose-Dilao, S., Orines, F. B., & Bernabe, J. G. (2003). Advanced Algebra, Trigonometry and Statistics. Quezon City, Philippines: JTW Corporation Lamayo, F. C., & Deauna, M. C. (1990). Fourth Year Integrated Mathematics. Quezon City, Philippines: Phoenix Publishing House, Inc. Larson, R., & Hostetler, R. P. (2012). Algebra and Trigonometry. Pasig City, Philippines: Cengage Learning Asia Pte Ltd Marasigan, J. A., Coronel, A. C., & Coronel, I. C. (2004). Advanced Algebra with Trigonometry and Statistics. Makati City, Philippines: The Bookmark, Inc. Quimpo, N. F. (2005). A Course in Freshman Algebra. Quezon City, Philippines Uy, F. B., & Ocampo, J. L. (2000). Board Primer in Mathematics. Mandaluyong City, Philippines: Capitol Publishing House. Villaluna, T. T., & Van Zandt, GE. L. (2009). Hands-on, Minds-on Activities in Mathematics IV. Quezon City, Philippines: St. Jude Thaddeus Publications.

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Module 4: Circles A. Learning Outcomes

Content Standard:

The learner demonstrates understanding of key concepts of circles. Performance Standard:

The learner is able to formulate and find solutions to challenging situations involving circles and other related terms in different disciplines through appropriate and accurate representations.


Subject: Mathematics 10 Quarter: Second Quarter TOPIC: Circles LESSONS: 1. A. Chords, Arcs, and Central Angles

B. Arcs and Inscribed Angles

2. A. Tangents and Secants

of a Circle B. Tangent and Secant Segments

Learning Competencies 1. Derive inductively the relations among

chords, arcs, central angles, and inscribed angles

2. Illustrate segments and sectors of circles 3. Prove theorems related to chords, arcs,

central angles, and inscribed angles 4. Solve problems involving chords, arcs,

central angles, and inscribed angles of circles

5. Illustrate tangents and secants of circles 6. Prove theorems on tangents and secants 7. Solve problems involving tangents and

secants of circles

Writer: Concepcion S. Ternida

Essential Understanding: Students will understand that the concept of circles has wide applications in real life and is a useful tool in problem-solving and in decision making.

Essential Question: How do geometric relationships involving circles help solve real-life problems that are circular in nature?

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Transfer Goal: Students will be able to apply the key concepts of circles in finding solutions and in making decisions for certain real-life problems.


Product/Performance

The following are products and performances that students are expected to come up with in this module.

1. Objects or situations in real life where chords, arcs, and central angles of circles are illustrated

2. A circle graph applying the knowledge of central angles, arcs, and sectors of a circle

3. Sketch plans or designs of a stage with circular objects that illustrate the use of inscribed angles and arcs of a circle

4. Sketch plans or designs of an arch bridge that illustrate the applications of secants and tangents

5. Deriving geometric relationships involving circles 6. Proof of theorems and other geometric relationships involving circles 7. Formulated and solved real-life problems

Assessment Map

TYPE KNOWLEDGE PROCESS/

SKILLS UNDERSTANDING PERFORMANCE


Pre-Test: Part I Identifying inscribed angle Identifying the external secant segment Describing the opposite angles of a quadrilateral inscribed in a circle Identifying the sum of the measures of the central angles of a circle

Pre-Test: Part I Finding the length of an arc of a circle given its radius Finding the measure of a central angle given its intercepted arc Finding the lengths of segments formed by intersecting chords

Pre-Test: Part I Part II Solving problems involving the key concepts of circles

118


Describing the inscribed angle intercepting a semicircle Determining the number of line that can be drawn tangent to the circle

Finding the measure of the angle formed by two secants Finding the length of a chord that is perpendicular to a radius Finding the length of a secant segment Finding the area of a sector of a circle Finding the measure of a central angle given its supplement Finding the measure of an angle of a quadrilateral inscribed in a circle Finding the measure of an inscribed angle given the measure of a central angle intercepting the same arc

Pre-Test: Part III Situational Analysis Planning the design of a garden

Pre-Test: Part III Situational Analysis Illustrating every part or portion of the garden including their measurements and accessories

Pre-Test: Part III Situational Analysis Explaining how to prepare the designs of the garden

Pre-Test: Part III Situational Analysis Making designs of gardens

119


Determining the mathematics concepts or principles involved in the design of the garden

Formulating problems that describe the situations Solving the problems formulated

Formative Quiz: Lesson 1A Identifying and describing terms related to circles

Quiz: Lesson 1A Solving the degree measure of the central angles and arcs Finding the length of the unknown segments in a circle Determining the reasons to support the given statements in a two-column proof of a theorem Solving the length of an arc of a circle given its degree measure Finding the area of the shaded region of circles

Quiz: Lesson 1A Justifying why angles or arcs are congruent Explaining why an arc is a semicircle Explaining how to find the degree measure of an arc Explaining how to find the center of a circular garden Solving real-life problems involving the chords, arcs, and central angles of circles

Quiz: Lesson 1B Identifying the inscribed angles and their intercepted arcs

Quiz: Lesson 1B Finding the measure of an inscribed angle and its intercepted arc

Quiz: Lesson 1B Explaining why the inscribed angles are congruent Proving theorems on inscribed

120


Determining the measure of an inscribed angle that intercepts a semicircle Determining the reasons to support the given statements in a two-column proof of a theorem

angles and intercepted arcs using two-column proofs Proving congruence of triangles using the theorems on inscribed angles Solving real-life problems involving arcs and inscribed angles Explaining the kind of parallelogram that can be inscribed in a circle

Quiz: Lesson 2A Identifying tangents and secants including the angles they form

Quiz: Lesson 2A Determining the measures of the different angles, arcs, and segments

Quiz: Lesson 2A Proving theorems on tangents and secants using two-column proofs Explaining how to find the measure of an angle given a circle with tangents Solving real-life problems involving tangents and secants of a circle

Quiz: Lesson 2B Identifying the external secant segment in a circle

Quiz: Lesson 2B Finding the length of the unknown segment in a circle

Quiz: Lesson 2B Proving theorems on intersecting chords, secant segments, and tangent segments Explaining why the solution for finding the length

121


Drawing a circle with appropriate labels and description

of a segment is correct or incorrect Solving real-life problems involving tangent and secant segments

Summative Pre-Test: Part I Identifying an inscribed angle Identifying a tangent Describing the angles of a quadrilateral inscribed in a circle Identifying the sum of the measures of the central angles of a circle Describing the inscribed angle intercepting a semicircle Determining the number of lines that can be drawn tangent to the circle

Pre-Test: Part I Finding the measure of an arc intercepted by a central angle Finding the length of an arc Finding the lengths of segments formed by intersecting chords Finding the measure of the angle formed by a tangent and a secant Finding the measure of an inscribed angle given the measure of a central angle intercepting the same arc Finding the length of a secant segment Finding the area of a sector of a circle

Pre-Test: Part I Part II Solving problems involving the key concepts of circles

Post-Test: Part III A and B Preparing sketches of the different formations to be followed in the field demonstrations including their sequencing and presentation on how each will be performed Formulating and solving problems involving the key concepts of circles

122


Finding the measure of a central angle given its supplement Finding the measure of an angle of a quadrilateral inscribed in a circle Finding the length of a chord that is perpendicular to a radius

Self-Assessment

Journal Writing: Expressing understanding of the key concepts of circles Expressing understanding of the different geometric relationships involving circles

Assessment Matrix (Summative Test)

Levels of Assessment What will I assess? How will I

assess? How Will I

Score?

Knowledge 15%

The learner demonstrates understanding of key concepts of circles. 1. Derive inductively the

relations among chords, arcs, central angles, and inscribed angles.

2. Illustrate segments and sectors of circles.

3. Prove theorems related to chords, arcs, central angles and inscribed angles

4. Solve problems involving chords, arcs, central angles, and inscribed angles of circles

Paper and Pencil Test Part I items 1, 3, 4, 6, 7, and 10


Process/Skills 25%

Part I items 2, 5, 8, 9, 11, 12, 13, 14, 15, and 16


Understanding 30%

Part I items 17, 18, 19, and 20


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5. Illustrate tangents and secants of circles

6. Prove theorems on tangents and secants

7. Solve problems involving tangents and secants of circles

Part II items 1 and 2

Rubric on Problem Solving (maximum of 4 points for each problem)


30%

The learner is able to formulate and find solutions to challenging situations involving circles and other related terms in different disciplines through appropriate and accurate representations.

Part III A Part III B

Rubric for Sketches of the Different Formations (Total Score: maximum of 6 points ) Rubric on Problems Formulated and Solved (Total Score: maximum of 6 points )


This module covers key concepts of circles. It is divided into four lessons namely: Chords, Arcs, and Central Angles, Arcs and Inscribed Angles, Tangents and Secants of a Circle, and Tangent and Secant Segments.

Lesson 1A is about the relations among chords, arcs and central

angles of a circle, area of a segment and a sector, and arc length of a circle. In this lesson, the students will determine the relationship between the measures of the central angle and its intercepted arc, apply the different geometric relationships among chords, arcs, and central angles in solving problems, complete the proof of a theorem related to these concepts, find the area of a segment and the sector of a circle, and determine the length of an arc. (Note that all measures of angles and arcs are in degrees.)

Moreover, the students will be given the opportunity to demonstrate

their understanding of the lesson by naming objects and citing real-life situations where chords, arcs, and central angles of a circle are illustrated and applied.

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The concepts about arcs and inscribed angles of a circle are contained in Lesson 1B. In this lesson, the students will determine the geometric relationships that exist among arcs and inscribed angles of a circle, apply these in solving problems, and prove related theorems. Moreover, they will formulate and solve real-life problems involving these geometric concepts.

The geometric relationships involving tangents and secants and

their applications in real life will be taken up in Lesson 2A. In this lesson, the students will find the measures of angles formed by secants and tangents and the arcs that these angles intercept. They will apply the relationships involving tangents and secants in finding the lengths of segments of some geometric figures. Moreover, the students will be given opportunities to formulate and solve real-life problems involving tangents and secants of a circle.

Lesson 2B of this module is about the different geometric

relationships involving tangent and secant segments. The students will apply these geometric relationships in finding the lengths of segments formed by tangents and secants. To demonstrate their understanding of the lesson, the students will make a design of a real-life object where tangent and secant segments are illustrated or applied, then formulate and solve problems out of this design.

In all the lessons, the students are given the opportunity to use their

prior knowledge and skills in learning circles. They are also given varied activities to process the knowledge and skills learned and further deepen and transfer their understanding of the different lessons.

As an introduction to the main lesson, show the students the

pictures below, then ask them the questions that follow:

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Entice the students to find out the answers to these questions and

to determine the vast applications of circles through this module. Objectives:

After the learners have gone through the lessons contained in this

module, they are expected to: 1. identify and describe terms related to circles; 2. use the relationship among chords, arcs, central angles, and

inscribed angles of circles;

3. find the area of segments and sectors of circles;

4. find the lengths of arcs of circles;

5. use two-column proofs in proving theorems related to chords, arcs, central angles, and inscribed angles of circles;

6. identify the tangents and secants of circles;

7. formulate and solve problems involving chords, arcs, central angles,

and inscribed angles of circles;

8. use two-column proofs in proving theorems related to tangents and secants of circles; and

9. formulate and solve problems involving tangents and secants of

circles.

PRE-ASSESSMENT:

Have you imagined yourself pushing a cart or riding a bus having wheels that are not round? Do you think you can move heavy objects from one place to another easily or travel distant places as fast as you can? What difficulty do you think would you experience without circles? Have you ever thought of the importance of circles in the field of transportation, industries, sports, navigation, carpentry, and in your daily life?

Check students’ prior knowledge, skills, and understanding of mathematics concepts related to circles. Assessing these will facilitate teaching and students’ understanding of the lessons in this module.

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Students are expected to demonstrate understanding of key concepts of circles, formulate real-life problems involving these concepts, and solve these using a variety of strategies. They are also expected to investigate mathematical relationships in various situations involving circles. Lesson 1A: Chords, arcs, and Central angles

What to Know

Assess students’ knowledge of the different mathematics concepts previously studied and their skills in performing mathematical operations. Assessing these will facilitate teaching and students’ understanding of chords, arcs, and central angles. Tell them that as they go through this lesson, they have to think of this important question: “How do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions?” Ask the students to identify, name, and describe the terms related to circles by doing Activity 1. Let them explain how they arrived at their answers. Also, ask them to describe and differentiate these terms.

Answer Key Part I Part II (Use the rubric to rate students’ 1. B 11. A works/outputs) 2. A 12. A 1. 24.67 m 3. D 13. B 2. 27.38 km 4. D 14. A 5. C 15. A Part III (Use the rubric to rate students’ 6. C 16. A works/outputs) 7. C 17. A 8. B 18. C 9. A 19. B

10. D 20. C

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Activity 1: Know My Terms and Conditions…

Answer Key 1. AN , AJ , AE 5. JL , JN , EN , EL 2. EJ 6. LEN , LJE , ENL , JLN , LNJ 3. EL , EJ 7. JAN , NAE 4. JNE , JLE 8. LEJ , JEN Questions:

a. Recall the definition of the terms related to circles.

Terms related to circle Description 1. radius It is a segment drawn from the center of

the circle to any point on the circle. 2. diameter It is a segment whose endpoints are on the

circle and it passes through the center of the circle. It is the longest chord.

3. chord It is a segment joining any two points on the circle.

4. semicircle It is an arc measuring one-half of the circumference of a circle.

5. minor arc It is an arc of a circle that measures less than a semicircle.

6. major arc It is an arc of a circle that measures greater than a semicircle.

7. central angle It is an angle whose vertex is at the center of the circle and with two radii as its sides.

8. inscribed angle It is an angle whose vertex is on a circle and whose sides contain chords of the circle.

128

Show the students the right triangles with different measures of sides

and let them find the missing side. Give focus on the mathematics concepts or principles applied to find the unknown side particularly the Pythagorean theorem. Activity 2: What is my missing side?

Answer Key

b. 1. A radius is half the measure of the diameter. 2. A diameter is twice the measure of the radius and it is the longest

chord. 3. A chord is a segment joining any two points on the circle. 4. A semicircle is an arc measuring one-half the circumference of a

circle. 5. A minor arc is an arc of a circle that measures less than the

semicircle. 6. A major arc is an arc of a circle that measures greater than the

semicircle. 7. A central angle is an angle whose vertex is the center of the circle

and with two radii as its sides. 8. An inscribed angle is an angle whose vertex is on a circle and

whose sides contain chords of the circle.

Answer Key

1. 10c units 2. 49.17c units 3. 73.12c units 4. 12a units 5. 4b units 6. 12.12b units

Questions:

a. Using the equation 222 cba . b. Pythagorean theorem

129

Provide the students with an opportunity to derive the relationship between the measures of the central angle and the measure of its intercepted arc. Ask them to perform Activity 3. In this activity, students will measure the angles of the given figures using a protractor. Ask them to get the sum of the angles in the first figure as well as the sum of the central angles in the second figure. Ask them also to identify the intercepted arc of each central angle. Emphasize that the sum of the angles formed by the coplanar rays with common vertex but with no common interior points is equal to the sum of the central angles formed by the radii of a circle with no common interior points. Activity 3: Measure Me and You Will See…

Answer Key

1. a. 105 d. 90 b. 75 e. 30 c. 60 2. a. 105 d. 90 b. 75 e. 30 c. 60 3. In each figure, the angles have a common vertex. 4. 360 ; 360 5. 360 6. 360 7.

Central Angle

Measure

Intercepted Arc

1. FAB

105

FB

2. BAC

75

BC

3. CAD

60

CD

4. EAD

90

ED

5. EAF

30

EF 8. 360 because the measure of the central angle is equal to the

measure of its intercepted arc. 9. Equal

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Present a real-life situation to the students to develop their understanding of arcs and central angles of circles. In this activity, ask them to find the degree measure of each arc of the wheel and also the angle formed at the hub. Ask them further the importance of the spokes of the wheel. Activity 4: Travel Safely

Before proceeding to the next activities, let the students give a brief summary of what they have learned so far. Provide them with an opportunity to relate or connect their responses in the activities given to this lesson. Let the students read and understand some important notes on chords, arcs, and central angles. Tell them to study carefully the examples given. What to PROCESS

In this section, let the students apply the key concepts of chords, arcs, and central angles. Tell them to use the mathematical ideas and the examples presented in the preceding section to answer the activities provided.

Ask the students to perform Activity 5. In this activity, the students will

identify and name arcs and central angles in the given circle and explain how they identified them.

Answer Key

a. 60 ; 60 b. Evaluate students’ responses

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Activity 5: Identify and Name Me

In activities 6, 7, and 8, ask the students to apply the different

geometric relationships in finding the degree measure of the central angles, the arcs that the angles intercept, and the lengths of chords. Then, let them explain how they arrived at their answers.

Answer Key

1. LMH (or LGH ) and LKH (or LJH );JKM (or JLM ) and

JGM (or JHM ) 2.

Minor Arcs

Major Arcs

JK

KMJ

KL

KGL

LM

LJM

MG

MKG

HG

HKG

JH

JMH

Note: There are many ways of naming the major arcs. The given answers are just some of those ways.

3. Some Possible Answers: LAM ; MAG ; GAH ; JAH ; JAK ;

LAK Questions:

a. A semicircle is an arc with measure equal to one-half of the circumference of a circle and is named by using the two endpoints and another point on the arc.

A minor arc is an arc of a circle that measures less than the semicircle. It is named by using the two endpoints on the circle.

A major arc is an arc of a circle that measures greater than the semicircle. It is named by using the two endpoints and another point on the arc.

A central angle is an angle whose vertex is the center of the circle and with two radii as its sides.

b. Yes. A circle has an infinite set of points. Therefore, a circle has many semicircles, arcs, and central angles.

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Activity 6: Find My Degree Measure

Activity 7: Find Me!

Activity 8: Get My Length

Provide the students opportunity to develop their skills in writing proofs. Ask them to complete the proof of a theorem involving the diameter, chord, and arc of a circle by doing Activity 9. If needed, guide the students as they complete the proof of the theorem.

Answer Key 1. 90 6. 90 2. 48 7. 48 3. 138 8. 150 4. 42 9. 42 5. 132 10. 132

Answer Key

1. JSO and NSI ; JSN and OSI . They are vertical angles. 2. a. 113

b. 67 c. 67

3. Yes. Yes. Opposite sides of rectangles are congruent. 4. JO and NI ;JN and OI . The central angles that intercept the arcs are congruent. 5. a. 67 d. 113

b. 113 e. 180 c. 67 f. 180

6. NJO ; NIO; JOI ;JNI . The arcs measure 180°. Each arc or semicircle contains the endpoints of the diameter.

Answer Key

1. 8 units 5. 24.639 units 2. 2 units 6. 8 units 3. 5 units 7. 29.572 units 4. 24.639 units 8. 581074 . units

Note: Evaluate students’ explanations.

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Activity 9: Make Me Complete! Problem: To prove that in a circle, a diameter bisects a chord and an arc

with the same endpoints if and only if it is perpendicular to the chord. The proof has two parts.

Given: ES is a diameter of U and perpendicular to chord GN at I. Prove: 1. GINI 2. EGEN 3. GSNS

N G

E

I

S

U

Answer Key Proof of Part 1: We will show that ES bisects GN and the minor arc GN.

Statements Reasons 1. U with diameter ES and chord

GN ; GNES Given

2. GIU and NIU are right angles. Definition of perpendicular lines 3. NIUGIU Right angles are congruent. 4. UNUG Radii of the same circle are

congruent.

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Answer Key Proof:

Statements Reasons 5. UIUI Reflexive/Identity Property 6. NIUGIU HyL Theorem 7. NIGI Corresponding parts of congruent

triangles are congruent (CPCTC).

8. ES bisects GN . Definition of segment bisector 9. NUIGUI From 6, CPCTC

10. GUI and GUE are the same angles.

NUI and NUE are the same angles.

E, I, U are collinear.

11. NUEmGUEm From 9, 10, definition of congruent angles

12. GUEmmEG

NUEmmEN

Degree measure of an arc

13. mEGmEN

From 11, 12, substitution

14. NUSmGUSm From 11, definition of supplementary angles, angles that are supplementary to congruent angles are congruent.

15. GUSmmGS

NUSmmNS

Degree measure of an arc

16. mGSmNS

From 14, 15, substitution

17. ES bisects GN .

Definition of arc bisector

135

Given: ES is a diameter of U; ES bisects GN

at I and the minor arc GN.

Combining Parts 1 and 2, the theorem is proven.

Have the students apply the knowledge and skills they have learned about arc length, segment, and sector of a circle. Ask the students to perform Activity 10 and Activity 11.

Answer Key Proof of Part 2: We will show that GNES .

Statements Reasons 1. U with diameter ES , ES

bisects GN at I and the minor arc

GN.

Given

2. NIGI

NEGE

Definition of bisector

3. UIUI Reflexive/Identity Property

4. UNUG Radii of the same circle are congruent.

5. NIUGIU SSS Postulate 6. UINUIG CPCTC 7. UIG and UIN are right

angles. Angles which form a linear pair and are congruent are right angles.

8. GNIU Definition of perpendicular lines

9. GNES IU is on ES

N G I

U

E

S

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Activity 10: Find My Arc Length

Activity 11: Find This Part!

Answer Key

1. 3.925 units 2. 32.5 units or 5.23 units 3. 7.85 units 4. 64.10 units or 10.47 units 5. 8.29 units

Questions:

a. The area of each shaded region was determined by using the

proportion r

lA

2360

where A = degree measure of the arc,

l = length of the arc, r = radius of the circle. Use the formula for finding the area of a segment and the area of a triangle.

b. The proportion r

lA

2360

, area of a segment and the area of a

triangle were used and so with substitution and the division property.

Answer Key

1. 9 cm2 or 28.26 cm2 2. 18 cm2 or 56.52 cm2 3. 52.77 cm2 4. 9.31 cm2 5. 59.04 cm2 6. 40 cm2

Questions:

a. The area of the sector is equal to the product of the ratio

360

arctheofmeasure and the area of the circle.

Subtract the area of the triangle from the area of the sector. b. Area of a circle, area of a triangle, ratio, equilateral triangle, and

regular pentagon

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What to REFLECT on and UNDERSTAND

Ask the students to take a closer look at some aspects of the geometric concepts contained in this lesson. Provide them opportunities to think deeply and test further their understanding of the lesson by doing Activity 12. In this activity, the students will solve problems involving chords, arcs, central angles, area of a segment and a sector, and arc length of a circle. Activity 12: More Circles Please …

Answer Key 1. a. 72

b. 3.768 cm c. regular pentagon

2. Yes. There are two pairs of congruent central angles/vertical angles formed and they intercept congruent arcs.

3. a. Yes. because the arcs are intercepted by the same central angle. b. No. Even if the two circles have the same central angles, the

lengths of their intercepted arcs are not equal because the 2 circles have different radii.

4. 60. (Evaluate students’ explanations. They are expected to use the

proportion r

lA

2360

to support their explanations.)

5. Draw two chords on the garden and a perpendicular bisector to each of the chords. The intersection of the perpendicular bisectors to the chord is the center of the circular garden.

6. a. Education, because it has the highest budget which is Php12,000.00 Savings & Utilities, because they have the lowest budget which

is Php4,500.00 b. Education. It should be given the greater allocation because it is

a very good investment. c. Education – 120 Food – 90 Utilities – 45 Savings – 45 Other expenses – 60 d. Get the percentage for each item by dividing the allotted budget by the monthly income, then multiply it by 360.

138

Before the students move to the next section of this lesson, give a short test (formative test) to find out how well they understood the lesson. Ask them also to write a journal about their understanding of chords, arcs, and central angles. Refer to the Assessment Map. What to TRANSFER

Give the students opportunities to demonstrate their understanding of circles by doing a practical task. Let them perform Activity 13. You can ask the students to work individually or in group. In this activity, the students will name 5 objects or cite 5 situations in real life where chords, arcs, and central angles of a circle are illustrated. Then, instruct them to formulate and solve problems out of these objects or situations. Also, ask them to make a circle graph showing the different school fees that students like them have to pay voluntarily like Parents-Teachers Association fee, miscellaneous fee, school paper fee, Supreme Student Government fee, and other fees. Ask them to explain how they applied their knowledge of central angles and arcs of circle in preparing the graph. Then, using the circle graph that they made, ask them to formulate and solve at least two problems involving arcs, central angles, and sectors of a circle. Activity 13: My Real World

Answer Key Evaluate students’ product. You may use the rubric provided.

e. Item Sector Arc Length

Education 61.654 cm2 52.3 cm Food 490.625 cm2 39.25 cm Utilities 245.3125 cm2 19.625 cm Savings 245.3125 cm2 19.625 cm Other expenses 308.327 cm2 61.26 cm

139

Summary/Synthesis/Generalization:

This lesson was about chords, arcs and central angles of a circle, area

of a segment and a sector, and arc length of a circle. In this lesson, the students determined the relationship between the measures of the central angle and its intercepted arc.

They were also given the opportunity to apply the different geometric

relationships among chords, arcs, and central angles in solving problems, complete the proof of a theorem related to these concepts, find the area of a segment and the sector of a circle, and determine the length of an arc.

Moreover, the students were asked to name objects and cite real-life

situations where chords, arcs, and central angles of a circle are illustrated and the relationships among these concepts are applied. Lesson 1B: Arcs and Inscribed Angles

What to KNOW

Let the students relate and connect previously learned mathematics concepts to the new lesson, arcs and inscribed angles. As they go through this lesson, tell them to think of this important question: “How do geometric relationships involving arcs and inscribed angles facilitate solving real-life problems and making decisions?”

Start the lesson by asking the students to perform Activity 1. In this

activity, the students will identify in a given figure the angles and their intercepted arcs. The students should be able to explain how they identified and named these angles and intercepted arcs.

140

Activity 1: My Angles and Intercepted Arcs

Give the students opportunity to determine the relationship between

the measure of an inscribed angle and the measure of its intercepted arc by performing Activity 2. The students should be able to realize in this activity that the measure of an angle inscribed in a circle is one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle).

Answer Key

Angles Arc That the Angle Intercepts

MSC

MC

CSD

CD

MSD

MD

MGC

MC

DGC

CD

MGD

MD

1. Determine the chords having a common endpoint on the circle. The chords are the sides of the angle and the common endpoint on the circle is the vertex.

Determine two radii of the circle. The two radii are the sides of the angle and the center of the circle is the vertex.

Determine the arc that lies in the interior of the angle with endpoints on the same angle.

2. There are 6 angles and there are also 6 arcs that these angles intercept.

3. An angle intercepts an arc if a point on one side of the angle is an endpoint of the arc.

141

Activity 2: Inscribe Me!

Answer Key Possible Responses

1. 2.

3. 60WELm ;

60mLW The measure of the central angle is equal to the measure of its intercepted arc.

4. 30LDWm

5. An inscribed angle is an angle whose vertex is on a circle and whose

sides contain chords of the circle.

6. The measure of LDW is one-half the measure of LW .

142

Activity 3 is related to Activity 2. In this activity, the students will determine the relationship that exists when an inscribed angle intercepts a semicircle. They should be able to find out that the measure of an inscribed angle that intercepts a semicircle is 90°. Activity 3: Intercept Me so I Won’t Fall!

Answer Key

1. 2.

3. 4.

5. a. 90MOTm b. 90MUTm c. 90MNTm

The measures of the three angles are equal. Each angle measures 90°. The measure of an inscribed angle intercepting a semicircle is 90°. The measures of inscribed angles intercepting the same arc are equal.

Answer Key

7. Draw other inscribed angles of the circle. Determine the measures of these angles and the degree measures of their respective intercepted arcs. (Check students’ drawings.)

The measure of an inscribed angle is one-half the degree measure of its intercepted arc.

If an angle is inscribed in a circle, then the measure of the angle

equals one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle).

143

Develop students’ understanding of the lesson by relating it to a real-life situation. Ask them to determine the mathematics concepts that they can apply to solve the problem presented in Activity 4. Activity 4: One, Two,…, Say Cheese!

Answer Key 1.

2. Relationship between the central angle or inscribed angle and the arc that the angle intercepts.

3. Go farther from the house until the entire house is seen on the eye piece or on the LCD screen viewer of the camera.

80°

40°

New location where Janel could photograph the entire

house with the telephoto lens

144

Before proceeding to the next section of this lesson, let the students give a brief summary of the activities done. Provide them with an opportunity to relate or connect their responses in the activities given to their new lesson, Arcs and Inscribed Angles. Let the students read and understand some important notes on the different geometric relationships involving arcs and inscribed angles and let them study carefully the examples given. What to PROCESS

Give the students opportunities to use the different geometric relationships involving arcs and inscribed angles, and the examples presented in the preceding section to perform the succeeding activities.

Ask the students to perform Activities 5, 6, and 7. In these activities,

they will identify the inscribed angles and their intercepted arcs, and apply the theorems pertaining to these geometric concepts and other mathematics concepts in finding their degree measures. Provide the students opportunities to explain their answers. Activity 5: Inscribe, Intercept, then Measure

Answer Key 1. LCA , LCE , ACE , ALC , CAE , CAL , LAE , and AEC 2.

a. CAL b. ACE c. LCE and LAE d. ALC and AEC

3. a. 281m d. 564 m g. 287 m b. 622 m e. 1245 m h. 628 m c. 623 m f. 566 m i. 629 m

4. a. 52mCL c. 52mAE

b. 128mAC d. 128mLE

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Activity 6: Half, Equal or Twice As?

Activity 7: Encircle Me!

Answer Key 1. BDCBAC and ABDACD . If inscribed angles intercept the

same arc, then the angles are congruent.

2. 108mCD

3. 48ACBm

4. a. 7x c. 38DCAm b. 38ABDm d. 76mAD

5. a. 5x c. 52mBC b. 26BDCm d. 26BACm

Answer Key 1. 4.

a. 150mOA a. 105TIAm

b. 50mOG b. 82FAIm c. 80GOAm d. 25GAOm

2. 5.

a. 65CARm a. 116mTM b. 557.ACRm b. 64mMA c. 557.ARCm c. 116mAE

d. 115mAC d. 32MEAm

e. 115mAR e. 58TAMm

3. a. 35RDMm b. 55DRMm c. 90DMRm

d. 110mDM

e. 180mRD

146

In Activity 8, ask the students to complete the proof of the theorem on inscribed angle and its intercepted arc. This activity would further develop their skills in writing proofs which they need in proving other geometric relationships.

Activity 8: Complete to Prove! Problem: To prove that if an angle is inscribed in a circle, then the

measure of the angle equals one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle).

Case 1: Given: PQR inscribed in S and PQ is a diameter.

Prove: PRPQR m2

1m

Draw RS and let xPQR m .

P R

Q

S

x

147


Provide the students with opportunities to think deeply and test further

their understanding of the lesson. Let them prove the different theorems on arcs and inscribed angles of a circle and other geometric relationships by performing Activity 9 and Activity 10. Moreover, ask the students to solve the problems in Activity 11 for them to realize the wide applications of the lesson in real life.

Answer Key

Statements Reasons 1. PQR inscribed in S

and PQ is a diameter. Given

2. RSQS Radii of a circle are congruent.

3. QRS is an isosceles . Definition of isosceles triangle 4. QRSPQR

The base angles of an isosceles triangle are congruent.

5. QRSPQR mm

The measures of congruent angles are equal.

6. xQRS m

Transitive Property

7. xPSR 2m

The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

8. PRPSR mm

The measure of a central angle is equal to the measure of its intercepted arc.

9. xPR 2m

Transitive Property

10. PQRPR m2m

Substitution

11. PRQRS m2

1m

Multiplication Property of Equality

148

Activity 9: Prove It or Else …!

Answer Key 1. Case 3

Given: SMC inscribed in A.

Prove: Cm2

1m SSMC

To prove: Draw diameter MP.

Answer Key 1. Case 2

Given: KLM inscribed in O.

Prove: KMKLM m2

1m

To prove: Draw diameter LN.

Proof:

Statements Reasons

KNKLN m2

1m and MNMLN m

2

1m

The measure of an inscribed angle is one-half the measure of its intercepted arc (Case 1).

MNKNMLNKLN m2

1m2

1mm or

MNKNMLNKLN mm2

1mm

Addition Property

KLMMLNKLN mmm Angle Addition Postulate

KMMNKN mmm Arc Addition Postulate

KMKLM m2

1m

Substitution

149

Proof: Statements Reasons

Sm2

1m PPMS and PCPMC m

2

1m

The measure of an inscribed angle is one-half the measure of its intercepted arc (Case 1).

PMCSMCPMS mmm or PMSPMCSMC mmm

Angle Addition Postulate

PCSCPS mmm or

PSPCSC mmm

Arc Addition Postulate

PSPCPMSPMC m2

1m2

1mm or

PSPCPMSPMC mm2

1mm

By Subtraction

Cm2

1m SSMC

Substitution

2. Given: In T, PR and AC are the intercepted arcs of PQR

and ABC , respectively.

ACPR

Prove: ABCPQR


ACPR

Given

ACPR mm

Congruent arcs have equal measures.

PRPQR m2

1m and

ACABC m2

1m

The measure of an inscribed angle is one-half the measure of its intercepted arc.

ACPQR m2

1m

Substitution

ABCPQR mm

Transitive Property

ABCPQR Angles with equal measures are congruent.

150

3. Given: In C, GML intercepts semicircle GEL.

Prove: GML is a right angle.


GML intercepts semicircle GEL. Given

180m GEL The degree measure of a semicircle is 180.

GELGML m2

1m


1802

1m GML or 90m GML

Substitution

GML is a right angle. Definition of right angle 4. Given: Quadrilateral WIND is inscribed in Y .

Prove: 1. W and N are supplementary.

2. I and D are supplementary. To prove: Draw WY , IY , NY , and DY . Proof:

Statements Reasons

360mmmm DYWNYDIYNWYI The sum of the measures of the central angles of a circle is 360.

WIWYI mm , INIYN mm ,

NDNYD mm , and DWDYW mm


360mmmm DWNDINWI

Substitution

360mm DWIDNI


151

Activity 10: Prove to Me if You Can!

Answer Key 1. Given: MT and AC are chords of D.

and ATMC , Prove: THACHM .

Proof

Statements Reasons 1. MT and AC are chords of

D and .MC AT Given

2. MCA , ATM , CMT , and CAT are inscribed angles.

Definition of inscribed angle

3. ATMMCA and CATCMT

Inscribed angles intercepting the same arc are congruent.

4. THACHM ASA Congruence Postulate

M

A

T D

C

H

Answer Key

Statements Reasons

DNIDWI m2

1m and DWIDNI m

2

1m The measure of an inscribed

angle is one-half the measure of its intercepted arc.

DWIDNIDNIDWI m2

1m2

1mm or

DWIDNIDNIDWI mm2

1mm

By Addition

3602

1mm DNIDWI or

180mm DNIDWI

Substitution

W and N are supplementary. Definition of supplementary angles

360mmmm DNIW

The sum of the measures of the angles of a quadrilateral is 360.

360180mm DI Substitution 180mm DI Addition Property

I and D are supplementary. Definition of supplementary angles

152

Answer Key 2. Given: Quadrilateral DRIV is inscribed in E.

RV is a diagonal that passes through the center of the circle.

IVDV Prove: RVIRVD Proof:

Statements Reasons 1. RV is a diagonal that

passes through the center of the circle

Given

2. RVRV Reflexive Property 3. VRIDRV Inscribed angles intercepting the

same arc are congruent.

4. RIV and RDV are semicircles.

Definition of semicircle

5. RDV and RIV are right angles.

Inscribed angle intercepting a semicircle measures 90°

6. RVD and RVI are right triangles.

Definition of right triangle

7. RVIRVD Hypotenuse-Angle Congruence Theorem

3. Given: In A, NESE and .SC NT

Prove: TNECSE

Proof:

Statements Reasons

1. NESE and NTSC Given

2. NESE and NTSC If two arcs are congruent, then the chords joined by their respective endpoints are also congruent.

3. mNEmSE and mNTmSC

Congruent arcs have equal measures.

4. mECmSCmSE and mETmNTmEN


E I

V

R

D

S

E N

T A

C

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Activity 11: Take Me to Your Real-World!

Answer Key 1. a. 72°

b. 36°. The measure of an inscribed angle is one-half the measure of its intercepted arc.

2. Rectangle. In a circle, there is only one chord that can be drawn

parallel and congruent to another chord in the same circle. Moreover, the diagonals of the parallelogram are also the diameters of the circle. Hence, each inscribed angle formed by the adjacent sides of the parallelogram intercepts a semicircle and measures 90°.

3. 38°. If EG is drawn, the viewing angles of Joanna, Clarissa, and Juliana intercept the same arc. Hence, the viewing angles of Joanna and Juliana measure the same as the viewing angle of Clarissa.

4. Mang Ador has to draw two inscribed angles on the circle such that

each measures 90°. Then, connect the other endpoints of the sides of each angle to form the diameter. The point of intersection of the two diameters is the center of the circle.

5. a. PQR is a right triangle.

b. The length of RS is the geometric mean of the lengths of PS and QS .

c. PS = 6 in.; QS = 2 in.; RS = 32 in. d. 34RT in. and 34MN in.

Answer Key

Statements Reasons

1. mETmEC Substitution

2. ETEC

Definition of Congruence

3. Draw chord CT . Definition of chord of a circle 4. ETCECT Inscribed angles intercepting

congruent arcs are congruent. 5. CET is an isosceles triangle. Definition of isosceles triangle. 6. TECE The legs of an isosceles triangle are

congruent. 7. TNECSE SSS Congruence Postulate

154

Before the students move to the next section of this lesson, give a short test (formative test) to find out how well they understood the lesson. Ask them also to write a journal about their understanding of arcs and inscribed angles. Refer to the Assessment Map. What to TRANSFER

Give the students opportunities to demonstrate their understanding of the geometric relationships involving arcs and inscribed angles. In Activity 12, ask the students to make a design of a stage where a special event will be held. Tell them to include in the design some circular objects that illustrate the use of inscribed angles and arcs of a circle, and explain how they applied these concepts in preparing the design. Then, ask them to formulate and solve problems out of the design they made. You can ask the students to work individually or in groups. Activity 12: How special is the event?


This lesson was about arcs and inscribed angles of a circle. In this

lesson, the students were given the opportunity to determine the geometric relationships that exist among arcs and inscribed angles of a circle, apply these in solving problems, and prove related theorems. Moreover, they were given the chance to formulate and solve real-life problems involving these geometric concepts out of the product they were asked to come up with as a demonstration of their understanding of the lesson.

Answer Key

Evaluate students’ product. You may use the given rubric.

155

Lesson 2A: Tangents and Secants of a Circle

What to KNOW

Assess students’ prior mathematical knowledge and skills that are related to tangents and secants of a circle. This would facilitate teaching and guide the students in understanding the different geometric relationships involving tangents and secants of a circle.

Start the lesson by asking the students to perform Activity 1. This

activity would lead them to some geometric relationships involving tangents and segments drawn from the center of the circle to the point of tangency. That is, the radius of a circle that is drawn to the point of tangency is perpendicular to the tangent line and is also the shortest segment.

Activity 1: Measure then Compare!

Answer Key 1. Use a compass to draw S.

2. Draw line m such that it intersects S at exactly one point. Label the point of intersection as T.

3. Connect S and T by a line segment. What is TS in the figure drawn?

TS is a radius of S.

156

4. Mark four other points on line m such that two of these points are on the left side of T and the other two points are on the right side. Label these points as M, N, P, and Q, respectively.

5. Using a protractor, find the measures of MTS , NTS , PTS, and

QTS . How do the measures of the four angles compare? The four angles have equal measures. Each angle measures 90°.

6. Repeat step 2 to 5. This time, draw line n such that it intersects the circle at another point. Name this point V. The four angles, AVS , BVS , DVS , and EVS have equal measures. Each angle measures 90°.

157

7. Draw MS , NS , PS , and QS .

8. Using a ruler, find the lengths of TS , MS , NS , PS , andQS .

How do the lengths of the five segments compare? The lengths of the five segments, TS , MS , NS , PS , and QS are not equal. What do you think is the shortest segment from the center of a circle to the line that intersects it at exactly one point? Explain your answer. The shortest segment from the center of a circle to the line that intersects the circle at exactly one point is the segment perpendicular to the line. Whereas, the other segments become the hypotenuses of the right triangles formed. Recall that the hypotenuse is the longest side of a right triangle.

158

Provide the students with opportunities to investigate relationships among arcs and angles formed by secants and tangents. Ask them to perform Activity 2 and Activity 3. Let the students realize the following geometric relationships:

1. If two secants intersect on a circle, then the measure of the angle formed

is one-half the measure of the intercepted arc. (Note: Relate this to the relationship between the measure of the inscribed angle and the measure of its intercepted arc.)

2. If a secant and a tangent intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

3. If a secant and a tangent intersect at the point of tangency, then the

measure of each angle formed is one-half the measure of its intercepted arc.

4. If two secants intersect in the exterior of a circle, then the measure of the

angle formed is one-half the positive difference of the measures of the intercepted arcs.

5. If two tangents intersect in the exterior of a circle, then the measure of the


6. If two secants intersect in the interior of a circle, then the measure of an

angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

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Activity 2: Investigate Me!

Answer Key

1. Which lines intersect circle C at two points? AD, AE, DG,

How about the lines that intersect the circle at exactly one point? BG

2. What are the angles having A as the vertex? DAG,DAB,EAG,DAE . There are still other angles with A as

the vertex, but for the purpose of our new lesson, we consider these angles. C as the vertex? DCE,ECF,ACG,ACD D as the vertex? .ADG There are still other angles with D as the vertex but for the meantime, we only consider this. G as the vertex?AGD. There are still other angles with G as the vertex but for the meantime, we only consider this.

3. DAB AD DCE DE

DAE DE ACD AD DAG DEA ACF AF EAG EFA ECF EF ADF AF AGD AF and AD

4. DAE and DCE DE

DAB , DCA , and AGD AD ACF , ADF , and AGD AF

5. 4334.DAEm 8768.ACGm

90EAGm 111.14m ECF 5755.DABm m DCE 68.87 43124.DAGm 4334.ADGm

111.14m ACD 1321.AGDm

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Activity 3: Find Out by Yourself!

Answer Key 6. Determine the measure of the central angle that intercepts the same

arc. The measure of the central angle is equal to the measure of its intercepted arc.

mAD= 111.14 mEFA= 180 mDE= 68.86 mEF= 111.14

mDEA= 248.86 mAF= 68.86

7. DAEmDCEm 2 DAEmmDE 2 . Since mDEDCEm ,

then DAEmmDE 2 .

8. DABmmAD 2 EAGmmEFA 2

9. mAFmADBGDm 21

Answer Key 2. RSTRST is a central angle of S.

4. mSTRSTm21

6. Yes. mRTmRVTRSTm 21

8. Yes. mNTmRTRSTm 21

10. Yes. mMNmRTRSTm 21

12. Yes. mMNmRTRSTm 21

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Let the students give their realizations of the activities done before proceeding to the next activities. Provide them with an opportunity to relate or connect their responses to the activities given in their lesson, tangents and secants of a circle. Let the students read and understand some important notes on tangents and secants of a circle and study carefully the examples given. What to PROCESS

In this section, let the students use the geometric concepts and relationships they have studied and the examples presented in the preceding section to answer the succeeding activities.

Present to the students the figure given in Activity 4. In this activity, the

students should be able to identify the tangents and secants in the figure including the angles that they form and the arcs that these angles intercept. They should be able to determine also the unknown measure of the angle formed by secants intersecting in the exterior of the circle. Give emphasis to the geometric relationship the students applied in finding the measure of the angle. Provide them opportunities to compare their answers and correct their errors, if there are any. Activity 4: Tangents or Secants?

Answer Key

1. KL and LM. Each line intersects the circle at exactly one point.

2. KN and MP. Each line intersects the circle at two points.

3. KNK and N; MPM and P; KLK; LMM

4. There are other angles formed but only these are considered.

KOM is formed by two secant lines. KLM is formed by two tangent lines.

LMP,LKN, PMR, and NKS. Each is formed by a secant and a tangent.

5. PMRMP , KOMNP , NKSKN , KLMKM , KLMKPM

6. 50KLMm ; mNP = 30

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In Activity 5, provide the students with opportunities to apply the different geometric relationships in finding the measures of the angles formed by tangents and secants and the arcs that these angles intercept. Let them also determine the lengths of segments tangent to circle/s and other segments drawn on a circle. Ask them to support their answers by stating the geometric relationships applied. Activity 5: From One Place to Another What to REFLECT on and UNDERSTAND

Let the students think deeply and test further their understanding of the

different geometric relationships involving tangents and secants of circles by doing Activity 6. In this activity, they will apply these geometric relationships in solving problems.

Answer Key 1. 40ABCm 7. 61PQOm 2. 40MQLm 119PQRm 3. 47PTRm 8. a. 125mPW

133RTSm b. 5.27RPWm 4. a. 10x c. 5.62PRWm

b. 65mCG d. 5.27WREm c. 55mAR e. 5.62WERm

5. 71mMC f. 5.62WERm 6. 854OR 9. 546PQ

24RS 10. a. 6x 24854 KS b. 19ST

c. 19RT d. 19AT

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Activity 6: Think of These Relationships Deeply!

Provide the students with opportunities to prove theorems involving

tangents and secants of circles. Let them perform Activity 7. Guide the students in writing the proof. If needed, provide hints.

Answer Key 1.

a. 90RONm ; 90RONm . The radius of a circle is perpendicular to a tangent line at the point of tangency.

b. NUDNRO c. 59NROm d. 41NDUm ; 131DUOm e. 5RO ; 12DN ; 36DU

NRO is not congruent to DUN . The lengths of their sides are not equal.

2. LU is tangent to I. SC is also tangent to I.

3. a. LIRL . If two segments from the same exterior point are tangent

to a circle, then the two segments are congruent. b. LTILTR by HyL Theorem. c. 38ILTm ; 52ITLm ; 52RTLm d. 26TL ; 24LI ; 16AL

4. a. 6SZ b. 3DZ c. 57.CX d. 57.CY If two segments from the same exterior point are tangent to a circle, then the two segments are congruent.

5. 555 m 6.

a. 55Pm 55Rm 55Sm

b. The angle that I will make with the lighthouse must be less than 55°.

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Activity 7: Is this true?

Answer Key 1. Given: AB is tangent to C at D.

Prove: CDAB To prove: a. DrawAC b. Assume AB is not perpendicular

toCD and ACAB Proof:

Statement Reason

AB is not perpendicular toCD and ACAB .

Assumption

E is a point on AD such that DADE 2

Ruler Postulate

AEDA Betweenness and Congruence of Segments

CADCAE Right angles are congruent. ACAC Reflexive Property

CEACDA SAS Congruence Postulate CECD CPCTC

CECD The lengths of congruent segments are equal.

D and E are on C. Definition of circle D and E are the points of intersection of tangent line AB and C is not true.

A tangent intersects the circle at exactly one point.

CDAB

Only one line can be drawn on a circle that is tangent to it at the point of tangency.

2. Given:RS is a radius of S. RSPQ

Prove: PQ is tangent to S at R. To prove: Draw QS .

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Answer Key Proof:

Statement Reason

RS is a radius of S and RSPQ .

Given QS >RS

The shortest segment from the center of a circle to a line tangent to it is the perpendicular segment.

Q is not on S.

No other point of a tangent line other than the point of tangency lies on a circle.

PQ is tangent to S at R. A tangent intersects the circle at exactly one point.

3. Given: EM and EL are tangent to

S at M and L, respectively. Prove: ELEM

To prove: Draw MS , LS , and ES . Proof:

Statement Reason

LSMS Radii of the same circle are congruent.

LSEL and MSEM . A line tangent to a circle is perpendicular to the radius.

ESES Reflexive Property

ESM ESL Hypotenuse-Leg Congruence Theorem

ELEM CPCTC

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4. a. Given: RS and TS are tangent to V at R and T, respectively,

and intersect at the exterior S.

Prove: TRTQRRST mm2

1m

To prove: Draw RV , TV , and SV .

Proof: Statement Reason

SVTSVR (Proven) 90mm RSVRVS and 90mm TSVTVS

Acute angles of a right triangle are complementary.

RVTTVSRVS mmm Angle Addition Postulate

m 90 90180 2

RVT x xx

Substitution

xTR 2180m


360mm TRTQR

The degree measure of a circle is 360.

xTQR 2180m Substitution and Addition Property of Equality

RSTTSVRSV mmm Angle Addition Postulate

x

xxTSVRSV

2

mm

By Substitution and Addition

xRST 2m Transitive Property

x

xxTRTQR

22

21802180mm

By Substitution and Subtraction

167

Answer Key

RSTTRTQR m2mm

By Substitution

TRTQRRST mm2

1m

Multiplication Property

b. Given:KL is tangent to O at K.

NL is a secant that passes through O at M and N. KL and NL intersect at the exterior point L.

Prove: MKNPKKLN mm2

1m

To prove: Draw KM , MO , and KO . Let xMKLm so that xMKOm 90 and xKMOm 90 .


NPKNMK m2

1m


NLKMKLNMK mmm

The measure of the exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

KMKOM mm


168

90mm MKOMKL The sum of the measures of complementary angles is 90.

180mmm KOMMKOKMO

The sum of the measures of the interior angle of a triangle is 180

xKOM 2m Addition Property

xKM 2m

Transitive Property

KMMKL mm2 or KMMKL m2

1m


MKLNMKKMNPK mmm2

1m

2

1

MKLNLKMKL mmm NLKm

By Subtraction

KMNPKNLK mm2

1m

By Substitution

c. Given: AC is a secant that passes

through T at A and B.

EC is a secant that passes through T at E and D. AC and EC intersect at the exterior point C.

Prove: BDAEACE mm2

1m

To prove: Draw AD and BE .

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Answer Key Proof:

Statement Reason

ACEDACADE mmm


AEADE m2

1m and

BDDAB m2

1m


DABADEBDAE mmm2

1m

2

1

By Subtraction

ACEDACADE mmm Addition Property

BDAEACE m2

1m

2

1m or

BDAEACE mm2

1m

Transitive Property

5. Given: PR and QS are secants intersecting in the interior of V at T. PS and QR are the intercepted arcs of PTS and QTR .

Prove: QRPSPTS mm2

1m

To prove: Draw RS .

170

Proof:

Statement Reason

PSPRS m2

1m and

QRQSR m2

1m


QSRPRSQTR mmm The measure of the exterior angle of a triangle is equal to the sum of the measures of its remote interior angles.

QRPSQTR m2

1m

2

1m or

QRPSQTR mm2

1m

Substitution

PTSQTR mm The measures of vertical angles are equal.

QRPSPTS mm2

1m

Transitive Property

6. Given: MP and LN are secant and tangent, respectively, and intersect at C at the point of tangency, M.

Prove: MPNMP m2

1m and

MKPLMP m2

1m

To prove: Draw OP and OM .

Let xNMP m so that xOMP 90m and xOPM 90m .

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Before the students move to the next section of this lesson, give a short test (formative test) to find out how well they understood the lesson. Ask them also to write a journal about their understanding of tangents and secants of a circle. Refer to the Assessment Map.

Answer Key Proof:

Statement Reason

MPMOP mm


90mm OMPNMP

The sum of the measures of complementary angles is 90.

180mmm MOPOPMOMP The sum of the measures of a triangle is 180.

xMOP 2m Addition Property

xMP 2m Transitive Property

NMPMP m2m Substitution

MPNMP m2

1m


360mm MKPMP The degree measure of a circle is 360.

xMKP 2360m By Substitution and Subtraction

xMKP 1802m By Factoring

xLMP 9090m or xLMP 180m

Angle Addition Postulate

LMPMKP m2m Substitution

MKPLMP m2

1m


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What to TRANSFER

Give the students opportunities to demonstrate their understanding of the different geometric relationships involving tangents and secants of circles by doing a practical task. Let them perform Activity 8. You can ask the students to work individually or in a group. In this activity, the students will formulate and solve problems involving tangents and secants of circles as illustrated in some real-life objects. Activity 8: My Real World


This lesson was about the geometric relationships involving tangents

and secants of a circle, the angles they form and the arcs that these angles intercept. The lesson provided the students with opportunities to derive geometric relationships involving radius of a circle drawn to the point of tangency, investigate relationships among arcs and angles formed by secants and tangents, and apply these in solving problems. Moreover, they were given the chance to prove the different theorems on tangents and secants and demonstrate their understanding of these concepts by doing a practical task. Their understanding of this lesson and other previously learned mathematics concepts and principles will facilitate their learning of the wide applications of circles in real life.

Answer Key

Evaluate students’ product. You may use the rubric provided.

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Lesson 2B: Tangent and Secant Segments

What to KNOW

Find out how much students have learned about the different mathematics concepts previously studied and their skills in performing mathematical operations. Checking these will facilitate teaching and students’ understanding of the geometric relationships involving tangent and secant segments. Tell them that as they go through this lesson, they have to think of this important question: How do geometric relationships involving tangent and secant segments facilitate solving real-life problems and making decisions?

Provide the students with opportunities to enhance further their skills in

finding solutions to mathematical sentences previously studied. Let them perform Activity1. In this activity, the students will solve linear and quadratic equations in one variable. These mathematical skills are prerequisites to learning the geometric relationships involving tangent and secant segments.

Ask the students to explain how they arrived at the solutions and how

they applied the mathematics concepts or principles in solving each mathematical sentence.

Activity 1: What is my value?

Present to the students the figure in Activity 2. Then, let them identify

the tangent and secant lines and the chords, name all the segments they can see, and describe a point in relation to the circle. This activity has something to do with the lesson. Let the students relate this to the succeeding activities.

Answer Key

1. 9x 6. 5x 2. 5x 7. 8x 3. 6x 8. 32x 4. 9x 9. 53x 5. 12x 10. 54x

Questions: a. Applying the Division Property of Equality and Extracting Square

Roots b. Division Property of Equality and Extracting Square Roots

174

Activity 2: My Segments

Ask the students to perform Activity 3 to determine the relationship that

exists among segments formed by intersecting chords of a circle. In this activity, the students might not be able to arrive at the accurate measurements of the chords due to the limitations of the measuring instrument to be used. If possible, use math freeware like GeoGebra in performing the activity.

Activity 3: What is true about my chords?

Present to the students a situation that would capture their interest and develop their understanding of the lesson. Let them perform Activity 4. In this activity, the students will determine the mathematics concepts or principles to solve the given problem.

Answer Key

1. JL - tangent; JS - secant; AS ; AT ; LN - chords 2. NE ; ET ; AE ; EL 3. AS ; AJ ; JL 4. A point outside the circle

Answer Key

1-2.

3. a. BA= 2.8 units c. MA = 1.95 units b. TA = 2.8 units d. NA= 4.02 units 4. The product of BAand TA is equal to the product of MA and NA . 5. If two chords of a circle intersect, then the product of the measures

of the segments of one chord is equal to the product of the measures of the segments of the other chord. (Emphasize this idea.)

175

Activity 4: Fly Me to Your World

Ask the students to summarize the activities done before proceeding to the next activities. Provide them with an opportunity to relate or connect their responses in the activities given to their new lesson, Tangent and Secant Segments. Let the students read and understand some important notes on tangent and secant segments and study carefully the examples given. What to PROCESS

Let the students use the different geometric relationships involving tangent and secant segments and the examples presented in the preceding section to answer the succeeding activities.

In Activity 5, the students will name the external secant segments in

the given figures. This activity would familiarize them with the geometric concept and facilitate problem solving. Activity 5: Am I away from you?

Answer Key

1. d = 27.67 km 2. External secant segment, tangent, Pythagorean theorem

Answer Key 1. IM and IL 2. TS and DS 3. OS 4. IR 5. LF and WE 6. IH , FG , IJ , EF , AK , DC

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Have the students apply the different theorems involving chords and tangent and secant segments to find the unknown lengths of segments on a circle and solve related problems. Ask the students to perform Activity 6 and Activity 7. Activity 6: Find My Length!

Activity 7: Try to Fit!

Answer Key

1. 8x units 6. 5.10x units 2. 8x units 7. 8.4x units 3. 9x units 8. 15x units 4. 5x units 9. 32.6102 x units 5. 64.6x units 10. 4x units

Questions: a. The theorems on two intersecting chords, secant segments, tangent

segments, and external secant segments were applied. b. Evaluate students’ responses.

Answer Key

1. Possible answer:

2. a. VU = 4.57 units b. XU = 8 units

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What to REFLECT on and UNDERSTAND Test further students’ understanding of the different geometric

relationships involving tangent and secant segments including chords by doing Activity 8 and Activity 9. Let the students prove the different theorems on intersecting chords, secant segments, tangent segments, and external secant segments and solve problems involving these concepts. Activity 8: Prove Me Right!

Answer Key 1. Given: AB and DE are chords of C intersecting at M.

Prove: EMDMBMAM To prove: Draw AE and BD .


mBEBAEm2

1 and

mBEBDEm2

1


BDEBAE Inscribed angles intercepting the same arc are congruent.

DMB~AME AA Similarity Theorem

DM

BM

AM

EM

Lengths of sides of similar triangles are proportional.

EMDMBMAM Multiplication Property

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Answer Key 2. Given: DP and DS are secant segments of T drawn from exterior point D.

Prove: DRDSDQDP To prove: Draw PR and QS .


RSQQPR and SRPPQS

Inscribed angles intercepting the same arc are congruent.

DRPDQS Supplements of congruent angles are congruent

DRP~DQS AA Similarity Theorem

DQ

DS

DR

DP


DRDSDQDP Multiplication Property 3. Given: KL and KM are tangent

and secant segments, respectively of O drawn

from exterior point K. KM intersects O at N.

Prove: 2KLKNKM

To prove: Draw LM and LN .

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Activity 9: Understand Me More …

Answer Key

1. Janel. She used the theorem “If two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.”

2. Gate 1 is 91.65 m from the main road. 3. a. The point of tangency of the two light balls from the ceiling is about

44.72 cm. b. Anton needs about 1967.53 cm of string.

Answer Key


mLNNLKm2

1 and

mLNLMNm2

1


LMNmNLKm Transitive Property

LMNNLK Angles with equal measures are congruent.

LMNmNLMmLNKm


NLKmNLMmLNKm Substitution NLKmNLMmKLMm Angle Addition Postulate

KLMmLNKm Transitive Property

KLMLNK Angles with equal measures are congruent.

LNM~MKL AA Similarity Theorem

KN

KL

KL

KM


2KLKNKM


180

Find out how well the students understood the lesson by giving a short test (formative test) before proceeding to the next section. Ask them also to write a journal about their understanding of tangent and secant segments. Refer to the Assessment Map.

What to TRANSFER

Give the students opportunities to demonstrate their understanding of tangent and secant segments including chords of a circle by doing a practical task. Let them perform Activity 10. You can ask the students to work individually or in a group.

In Activity 10, the students will make a design of an arch bridge that

would connect two places which are separated by a river, 20 m wide. Tell them to indicate on the design the different measurements of the parts of the bridge. The students are expected to formulate and solve problems involving tangent and secant segments out of the design and the measurements of its parts.

Activity 10: My True World!


This lesson was about the different geometric relationships involving

tangents, secants, and chords of a circle. The lesson provided the students with opportunities to derive geometric relationship involving intersecting chords, identify tangent and secant segments, and prove and apply different theorems on chords, tangent, and secant segments. These theorems were used to solve various geometric problems. Understanding the ideas presented in this lesson will facilitate their learning of the succeeding lessons.

Answer Key

Evaluate students’ product. You may use the rubric provided.

181

SUMMATIVE TEST Part I Choose the letter that you think best answers each of the following questions. 1. In the figure on the right, which is an inscribed angle?

A. RST B. PQR C. QVT D. QST

2. In F below, AG is a diameter. What is mAD if 65DFGm ? A. 65° B. 115° C. 130° D. 230°

3. Which of the following lines is tangent to F as shown in the figure

below? A. DE

B. AG

C. BD

D. AE

182

4. Quadrilateral ABCD is inscribed in a circle. Which of the following is true about the angle measures of the quadrilateral?

I. 180 CmAm II. 180 DmBm III. 90 CmAm

A. I and II B. I and III C. II and III D. I, II, and III

5. An arc of a circle measures 72°. If the radius of the circle is 6 cm, about

how long is the arc? A. 1.884 cm B. 2.4 cm C. 3.768 cm D. 7.54 cm

6. What is the total measure of the central angles of a circle with no common

interior points? A. 480 B. 360 C. 180 D. 120

7. What kind of angle is the inscribed angle that intercepts a semicircle?

A. straight B. obtuse C. right D. acute

8. What is the length of AS in the figure on the right?

A. 6.92 units C. 14.4 units

B. 11710

units D. 1309

units

9. Line AB is tangent to C at D. If mDF = 166 and mDE = 78, what is

ABFm ? A. 44 B. 61 C. 88 D. 122

183

P E

U

R

H

A

E

L T

O

T

H 8 cm

45°

10. How many line/s can be drawn through a given point on a circle that is tangent to the circle? A. four B. three C. two D. one

11. In U on the right, what is PREm if 56PUEm ?

A. 28 C. 56 B. 34 D. 124

12. In the figure below, TA and HA are secants. If TA = 18 cm, LA = 8 cm,

and AE = 10 cm,

what is the length of AH in the given figure?

A. 18 cm C. 22.5 cm B. 20 cm D. 24.5 cm

13. In O on the right, mHT = 45 and the length of the

radius is 8 cm. What is the area of the shaded region in terms of ? A. 6 cm 2 C. 10 cm 2 B. 8cm 2 D. 12 cm 2

184

C

K

L

77° U

96°

S

14. In the circle on the right, what is the measure of SRT if AST is a semicircle and ?74SRAm A. 16 B. 74 C. 106 D. 154

15. Quadrilateral LUCK is inscribed in S. If 96m LUC and

,77m UCK find ULKm . A. 77

B. 84 C. 96 D. 103

16. In S on the right, what is RT if QS = 18 units

and VW = 4 units? A. 24 units B. 28 units C. 14 units D. 216 units

17. A circular garden has a radius of 2 m. Find the area of the smaller segment of the garden determined by a 90 arc. A. 2 m2 B. 2 m2 C. m

2 D. 24 m2

R A

S

T

185

18.

A. 60° B. 75° C. 120° D. 150°

19. Mang Jose cut a circular board with a diameter 80 cm. Then, he divided

the board into 20 congruent sectors. What is the area of each sector? A. 80 cm2 B. 320 cm2 C. 800 cm2 D. 6001 cm2

20. Mary designed a pendant. It is a regular octagon set in a circle. Suppose

the opposite vertices are connected by line segments and meet at the center of the circle. What is the measure of each angle formed at the center? A. 5.22 B. 45 C. 567. D. 135

Part II Solve each of the following problems. Show your complete solutions. 1. Mr. Jaena designed an arch for the top part of a subdivision’s main gate.

The arch will be made out of bent iron. In the design, the 16 segments between the two concentric semicircles are each 0.7 meter long. Suppose the diameter of the outer semicircle is 8 meters. What is the length, in whole meters, of the shortest iron needed to make the arch?

2. A rope fits tightly around two pulleys. What is the distance between the centers of the pulleys if the radii of the bigger and smaller pulleys are 10 cm and 6 cm, respectively, and the portion of the rope tangent to the two pulleys is 50 cm long?

30°

Karen has a necklace with a circular pendant hanging from a chain around her neck. The chain is tangent to the pendant. If the chain is extended as shown in the diagram on the right, it forms an angle of 30° below the pendant. What is the measure of the arc at the bottom of the pendant?

186

Rubric for Problem Solving

4 3 2 1 Used an appropriate strategy to come up with a correct solution and arrived at a correct answer

Used an appropriate strategy to come up with a solution, but a part of the solution led to an incorrect answer

Used an appropriate strategy but came up with an entirely wrong solution that led to an incorrect answer

Attempted to solve the problem but used an inappropriate strategy that led to a wrong solution

Part III A: GRASPS Assessment Perform the following. Goal: To prepare the different student formations to be done during a

field demonstration Role: Student assigned to prepare the different formations to be

followed in the field demonstration Audience: The school principal, your teacher, and your fellow students Situation: Your school has been selected by the municipal/city

government to perform a field demonstration as part of a big local event where many visitors and spectators are expected to arrive and witness the said occasion. The principal of your school designated one of your teachers to organize and lead the group of students who will perform the field demonstration.

Being one of the students selected to perform during the

activity, your teacher asked you to plan the different student formations for the field demonstration. In particular, your teacher instructed you to include arrangements that show geometric figures such as circles, arcs, tangents, and secants. Your teacher also asked you to make a sketch of the various formations and include the order in which these will be performed by the group.

187

Products: Sketches of the different formations to be followed in the field demonstrations including the order and manner on how each will be performed

Standards: The sketches of the different formations must be accurate and

presentable, and the sequencing must also be systematic. Rubric for Sketches of the Different Formations

4 3 2 1 The sketches of the different formations are accurately made, presentable, and the sequencing is systematic.

The sketches of the different formations are accurately made and the sequencing is systematic but not presentable.

The sketches of the different formations are not accurately made but the sequencing is systematic.

The sketches of the different formations are made but not accurate and the sequencing is not systematic.

Part III B Use the prepared sketches of the different formations in Part III A in formulating problems involving circles, then solve.

188

Rubric on Problems Formulated and Solved

Score Descriptors

6

Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate.

5

Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

4

Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

3

Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details.

2 Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension.

1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach.

Source: D.O. #73 s. 2012

Answer Key Part I Part II (Use the rubric to rate students’ works/outputs) 1. B 11. A 1. 35 m 2. B 12. C 2. 50.16 cm 3. D 13. B 4. A 14. C 5. D 15. D Part III A (Use the rubric to rate students’ works/outputs) 6. B 16. D Part III B (Use the rubric to rate students’ works/outputs) 7. C 17. A 8. D 18. D 9. A 19. A

10. D 20. B

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GLOSSARY OF TERMS Arc – a part of a circle

Arc Length – the length of an arc which can be determined by using the

proportion r

lA =360 2

, where A is the degree measure of an arc, r is the

radius of the circle, and l is the arc length

Central Angle – an angle formed by two rays whose vertex is the center of the circle Common External Tangents – tangents which do not intersect the segment joining the centers of the two circles

Common Internal Tangents – tangents that intersect the segment joining the centers of the two circles

Common Tangent – a line that is tangent to two circles on the same plane

Congruent Arcs – arcs of the same circle or of congruent circles with equal measures

Congruent Circles – circles with congruent radii

Degree Measure of a Major Arc – the measure of a major arc that is equal to 360 minus the measure of the minor arc with the same endpoints.

Degree Measure of a Minor Arc – the measure of the central angle which intercepts the arc

External Secant Segment – the part of a secant segment that is outside a circle Inscribed Angle – an angle whose vertex is on a circle and whose sides contain chords of the circle

Intercepted Arc – an arc that lies in the interior of an inscribed angle and has endpoints on the angle

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Major Arc – an arc of a circle whose measure is greater than that of a semicircle

Minor Arc – an arc of a circle whose measure is less than that of a semicircle Point of Tangency – the point of intersection of the tangent line and the circle

Secant – a line that intersects a circle at exactly two points. A secant contains a chord of a circle

Sector of a Circle – the region bounded by an arc of the circle and the two radii to the endpoints of the arc

Segment of a Circle – the region bounded by an arc and a segment joining its endpoints

Semicircle – an arc measuring one-half the circumference of a circle

Tangent to a Circle – a line coplanar with the circle and intersects it at one and only one point

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List of Theorems And Postulates On Circles Postulates: 1. Arc Addition Postulate. The measure of an arc formed by two adjacent

arcs is the sum of the measures of the two arcs.

2. At a given point on a circle, one and only one line can be drawn that is tangent to the circle.

Theorems: 1. In a circle or in congruent circles, two minor arcs are congruent if and only

if their corresponding central angles are congruent.

2. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

3. In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is perpendicular to the chord.

4. If an angle is inscribed in a circle, then the measure of the angle equals

one-half the measure of its intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle).

5. If two inscribed angles of a circle (or congruent circles) intercept congruent

arcs or the same arc, then the angles are congruent.

6. If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.

7. If a quadrilateral is inscribed in a circle, then its opposite angles are

supplementary.

8. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.

9. If a line is perpendicular to a radius of a circle at its endpoint that is on the

circle, then the line is tangent to the circle.

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10. If two segments from the same exterior point are tangent to a circle, then the two segments are congruent.

11. If two secants intersect in the exterior of a circle, then the measure of the


12. If a secant and a tangent intersect in the exterior of a circle, then the

measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs.

13. If two tangents intersect in the exterior of a circle, then the measure of

the angle formed is one-half the positive difference of the measures of the intercepted arcs.

14. If two secants intersect in the interior of a circle, then the measure of an

angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

15. If a secant and a tangent intersect at the point of tangency, then the

measure of each angle formed is one-half the measure of its intercepted arc.

16. If two chords of a circle intersect, then the product of the measures of

the segments of one chord is equal to the product of the measures of the segments of the other chord.

17. If two secant segments are drawn to a circle from an exterior point, then

the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.

18. If a tangent segment and a secant segment are drawn to a circle from an

exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.

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DEPED INSTRUCTIONAL MATERIALS THAT CAN BE USED AS ADDITIONAL RESOURCES FOR THE LESSON ON CIRCLES: 1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third

Year Mathematics. Module 18: Circles and Their Properties.

2. Distance Learning Module (DLM) 3, Module 1 and 2: Circles. References And Website Links Used in This Module: References: Bass, L. E., Charles, R.I., Hall, B., Johnson, A., & Kennedy, D. (2008). Texas

Geometry. Boston, Massachusetts: Pearson Prentice Hall. Bass, L. E., Hall B.R., Johnson A., & Wood, D.F. (1998). Prentice Hall

Geometry Tools for a Changing World. NJ, USA: Prentice-Hall, Inc. Boyd, C., Malloy, C., & Flores. (2008). McGraw-Hill Geometry. USA: The

McGraw-Hill Companies, Inc. Callanta, M. M. (2002). Infinity, Worktext in Mathematics III. Makati City:

EUREKA Scholastic Publishing, Inc. Chapin, I., Landau, M. & McCracken. (1997). Prentice Hall Middle Grades

Math, Tools for Success. Upper Saddle River, New Jersey: Prentice-Hall, Inc.

Cifarelli, V. (2009) cK-12 Geometry, Flexbook Next Generation Textbooks.

USA: Creative Commons Attribution-Share Alike. Clemens, S. R., O’Daffer, P. G., Cooney, T.J., & Dossey, J. A. (1990).

Geometry. USA: Addison-Wesley Publishing Company, Inc. Clements, D. H., Jones, K.W., Moseley, L. G., & Schulman, L. (1999). Math in

My World. Farmington, New York: McGraw-Hill Division. Department of Education. (2012) K to 12 Curriculum Guide Mathematics.

Department of Education, Philippines. Gantert, A. X. (2008) AMSCO’s Geometry. NY, USA: AMSCO School

Publications, Inc. Renfro, F. L. (1992) Addison-Wesley Geometry Teacher’s Edition. USA:

Addison-Wesley Publishing Company, Inc.

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Rich, B. and Thomas, C. (2009). Schaum’s Outlines Geometry (4th ed.) USA:

The McGraw-Hill Companies, Inc. Smith, S. A., Nelson, C.W., Koss, R. K., Keedy, M. L., & Bittinger, M. L.

(1992) Addison-Wesley Informal Geometry. USA: Addison-Wesley Publishing Company, Inc.

Wilson, P. S. (1993) Mathematics, Applications and Connections, Course I.,

Westerville, Ohio: Glencoe Division of Macmillan/McGraw-Hill Publishing Company.

Website Links as References and Source of for Learning Activities: CK-12 Foundation. cK-12 Inscribed Angles. (2014). Retrieved from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.7/ CK-12 Foundation. cK-12 Secant Lines to Circles. (2014). Retrieved from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.8/ CK-12 Foundation. cK-12 Tangent Lines to Circles. (2014). Retrieved from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.4/ Houghton Mifflin Harcourt. Cliffs Notes. Arcs and Inscribed Angles. (2013). Retrieved from http://www.cliffsnotes.com/math/geometry/circles/arcs-and-inscribed-angles Houghton Mifflin Harcourt. Cliffs Notes. Segments of Chords, Secants, and Tangents. (2013). Retrieved from http://www.cliffsnotes.com/math/geometry/circles/segments-of-chords-secants-tangents Math Open Reference. Arc. (2009). Retrieved from http://www.mathopenref.com/arc.html Math Open Reference. Arc Length. (2009). Retrieved from http://www.mathopenref.com/arclength.html Math Open Reference. Central Angle. (2009). Retrieved from http://www.mathopenref.com/circlecentral.html Math Open Reference. Central Angle Theorem. (2009). Retrieved from http://www.mathopenref.com/arccentralangletheorem.html

http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/%20section/8.7/



http://www.cliffsnotes.com/

http://www.cliffsnotes.com/

http://www.mathopenref.com/arclength.html

http://www.mathopenref.com/arccentralangletheorem.html

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Math Open Reference. Chord. (2009). Retrieved from http://www.mathopenref.com/chord.html Math Open Reference. Inscribed Angle. (2009). Retrieved from http://www.mathopenref.com/circleinscribed.html Math Open Reference. Intersecting Secants Theorem. (2009). Retrieved from http://www.mathopenref.com/secantsintersecting.html Math Open Reference. Sector. (2009). Retrieved from http://www.mathopenref.com/arcsector.html Math Open Reference. Segment. (2009). Retrieved from http://www.mathopenref.com/segment.html math-worksheet.org. Free Math Worksheets. Arc Length and Sector Area. (2014). Retrieved from http://www.math-worksheet.org/arc-length-and-sector-area math-worksheet.org. Free Math Worksheets. Inscribed Angles. (2014). Retrieved from http://www.math-worksheet.org/inscribed-angles math-worksheet.org. Free Math Worksheets. Secant-Tangent Angles. (2014). Retrieved from http://www.math-worksheet.org/secant-tangent-angles math-worksheet.org. Free Math Worksheets. Tangents. (2014). Retrieved from tangents OnlineMathLearning.com. Circle Theorems. (2013). Retrieved from http://www.onlinemathlearning.com/circle-theorems.html Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Formulas for Angles in Circles Formed by Radii, Chords, Tangents, Secants. (2012). Retrieved from http://www.regentsprep.org/Regents/math/geometry/ GP15/CircleAngles.htm

http://www.mathopenref.com/circleinscribed.html

http://www.mathopenref.com/secantsintersecting.html

http://www.mathopenref.com/arcsector.html

http://www.mathopenref.com/segment.html

http://www.math-worksheet.org/arc-length-and-sector-area

http://www.math-worksheet.org/arc-length-and-sector-area

http://www.math-worksheet.org/inscribed-angles

http://www.math-worksheet.org/secant-tangent-angles

http://www.onlinemathlearning.com/circle-theorems.html

http://www.regentsprep.org/Regents/math/geometry/%20GP15/CircleAngles.htm

196

Website Links for Videos: Coach, Learn. NCEA Maths Level 1 Geometric reasoning: Angles Within Circles. (2012). Retrieved from http://www.youtube.com/watch?v=jUAHw-JIobc Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved from https://www.khanacademy.org/math/geometry/cc-geometry-circles Schmidt, Larry. Angles and Arcs Formed by Tangents, Secants, and Chords. (2013). Retrieved from http://www.youtube.com/watch?v=I-RyXI7h1bM Sophia.org. Geometry. Circles. (2014). Retrieved from http://www.sophia.org/topics/circles Website Links for Images: Cherry Valley Nursery and Landscape Supply. Seasonal Colors Flowers and Plants. (2014). Retrieved from http://www.cherryvalleynursery.com/ eBay Inc. Commodore Holden CSA Mullins pursuit mag wheel 17 inch genuine - 4blok #34. (2014). Retrieved from http://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag-wheel-17-inch-genuine-4blok-34-/221275049465 Fort Worth Weekly. Facebook Fact: Cowboys Are World’s Team. (2012) . Retrieved from http://www.fwweekly.com/2012/08/21/facebook-fact-cowboys-now-worlds-team/ GlobalMotion Media Inc. Circular Quay, Sydney Harbour to Historic Hunter's Hill Photos. (2013). Retrieved from http://www.everytrail.com/ guide/circular-quay-sydney-harbour-to-historic-hunters-hill/photos HiSupplier.com Online Inc. Shandong Sun Paper Industry Joint Stock Co.,Ltd. Retrieved from http://pappapers.en.hisupplier.com/product-66751-Art-Boards.html Kable. Slip-Sliding Away. (2014). Retrieved from http://www.offshore-technology.com/features/feature1674/feature1674-5.html Materia Geek. Nikon D500 presentada officialmente. (2009). Retrieved from http://materiageek.com/2009/04/nikon-d5000-presentada-oficialmente/

http://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag-wheel-17-inch-genuine-4blok-34-/221275049465

http://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag-wheel-17-inch-genuine-4blok-34-/221275049465

http://www.globalmotion.com/

http://www.everytrail.com/

http://www.offshore-technology.com/features/feature1674/feature1674-5.html

http://www.offshore-technology.com/features/feature1674/feature1674-5.html

http://materiageek.com/2009/04/nikon-d5000-presentada-oficialmente/

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Piatt, Andy. Dreamstime.com. Rainbow Stripe Hot Air Balloon. Retrieved from http://thumbs.dreamstime.com/z/rainbow-stripe-hot-air-balloon-788611.jpg Regents of the University of Colorado. Nautical Navigation. (2014). Retrieved from http://www.teachengineering.org/view_activity.php?url= collection/cub_/activities/cub_navigation/cub_navigation_lesson07_activity1.xml Sambhav Transmission. Industrial Pulleys. Retrieved from http://www.indiamart.com/sambhav-transmission/industrial-pulleys.html shadefxcanopies.com. Flower Picture Gallery, Garden Pergola Canopies. Retrieved from http://www.flowerpicturegallery.com/v/halifax-public-gardens/Circular+mini+garden+with+white+red+flowers+and+dark+ grass+in+the+middle+at+Halifax+Public+Gardens.jpg.html Tidwell, Jen. Home Sweet House. (2012). Retrieved from http://youveneverheardofjentidwell.com/2012/03/02/home-sweet-house/ Weston Digital Services. FWR Motorcycles LTD. CHAINS AND SPROCKETS. (2014). Retrieved from http://fwrm.co.uk/index.php?main_page=index&cPath=585&zenid=10omr4hehmnbkktbl94th0mlp6

http://thumbs.dreamstime.com/z/rainbow-stripe-hot-air-balloon-788611.jpg

http://www.teachengineering.org/view_activity.php?url

http://www.flowerpicturegallery.com/v/halifax-public-gardens/Circular+mini+garden+with+white+red+flowers+and+dark

http://www.flowerpicturegallery.com/v/halifax-public-gardens/Circular+mini+garden+with+white+red+flowers+and+dark

http://youveneverheardofjentidwell.com/2012/03/02/home-sweet-house/

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Module 5: Plane Coordinate Geometry A. Learning Outcomes

Content Standard:

The learner demonstrates understanding of key concepts of coordinate geometry. Performance Standard:

The learner is able to formulate and solve problems involving geometric figures on the rectangular coordinate plane with perseverance and accuracy.


Subject: Mathematics 10 Quarter: Second Quarter Topic: Plane Coordinate Geometry Lessons: 1. The Distance

Formula 2. The Equation of a

Circle

Learning Competencies

Derive the distance formula Apply the distance formula to prove some geometric

properties Illustrate the center-radius form of the equation of a

circle Determine the center and radius of a circle given its

equation and vice versa Graph a circle and other geometric figures on the

coordinate plane Solve problems involving geometric figures on the

coordinate plane

Writer: Melvin M. Callanta

Essential Understanding: Students will understand that the concepts involving plane coordinate geometry are useful tools in solving real-life problems like finding locations, distances, mapping, etc.

Essential Question: How do the key concepts of plane coordinate geometry facilitate finding solutions to real-life problems involving geometric figures?

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Transfer Goal: Students will be able to apply with perseverance and accuracy the key concepts of plane coordinate geometry in formulating and solving problems involving geometric figures on the rectangular coordinate plane.


Product/Performance

The following are products and performances that students are expected to come up with in this module. 1. Ground Plan drawn on a grid with coordinates 2. Equations and problems involving mathematics concepts already learned

such as coordinate plane, slope and equation of a line, parallel and perpendicular lines, polygons, distance, angles, etc

3. Finding the distance between a pair of points on the coordinate plane 4. Determining the missing coordinates of the endpoints of a segment 5. Finding the coordinates of the midpoint of the segment whose endpoints

are given 6. Describing the figure formed by a set of points on a coordinate plane 7. Determining the missing coordinates corresponding to the vertices of

some polygons 8. Solutions to problems involving the distance and the midpoint formulas 9. Coordinate Proofs of some geometric properties 10. Sketch of a municipal, city, or provincial map on a coordinate plane with

the coordinates of some important landmarks 11. Formulating and solving real-life problems involving the distance and the

midpoint formula 12. Finding the radius of a circle drawn on a coordinate plane 13. Determining the center and the radius of a circle given the equation 14. Graphing a circle given the equation 15. Writing the equation of a circle given the center and the radius 16. Writing the equation of a circle from standard form to general form and

vice-versa 17. Determining the equation that describes a circle 18. Solutions to problems involving the equation of a circle 19. Formulating and solving real-life problems involving the equation of a

circle

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Assessment Map

TYPE KNOWLEDGE PROCESS/

SKILLS UNDERSTANDING PERFORMANCE


Pre-Test: Part I Identifying the distance formula Illustrating the distance between two points on the coordinate plane Illustrating the midpoint formula Illustrating the midpoint of a segment Defining coordinate proof Identifying an equation of a circle

Pre-Test: Part I Determining the distance between a pair of points Determining the coordinate of a point given its distance from another point Determining the coordinates of the midpoint and the endpoints of a segment Describing the figure formed by a set of points Determining the coordinates of the vertex of a geometric figure Finding the length of the radius of a circle given the endpoints of a diameter Finding the center of a circle given the equation Finding the equation of a circle given the endpoints of a radius

Pre-Test: Part I and Part II Solving problems involving the Distance Formula including the Midpoint Formula, and the Equation of a Circle

201


Pre-Test:

Part III Situational Analysis Determining the mathematics concepts or principles involved in a prepared ground plan

Pre-Test: Part III Situational Analysis Illustrating the locations of objects or groups Writing the equations that describe the situations or problems Solving equations

Pre-Test: Part III Situational Analysis Explaining how to prepare the ground plan for the Boy Scouts Jamboree Solving real-life problems

Pre-Test: Part III Situational Analysis Making a ground plan for the Boy Scouts Jamboree Formulating equations, inequalities, and problems

Formative Quiz: Lesson 1 Identifying the coordinates of points to be substituted in the distance formula and in the midpoint formula Identifying the figures formed by some sets of points Identifying parts of some geometric figures and their properties

Quiz: Lesson 1 Finding the distance between each pair of points on the coordinate plane Finding the coordinates of the midpoint of a segment given the endpoints Plotting some sets of points on the coordinate plane Naming the missing coordinates of the vertices of some geometric figures

Quiz: Lesson 1 Explaining how to find the distance between two points Explaining how to find the midpoint of a segment Describing figures formed by some sets of points Explaining how to find the missing coordinates of some geometric figures Solving real-life problems involving the distance formula and the midpoint formula Using coordinate proof to justify claims

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Writing a coordinate proof to prove geometric properties

Quiz: Lesson 2 Identifying the equations of circles in center-radius form or standard form and in general form

Quiz: Lesson 2 Determining the center and the radius of a circle Graphing a circle given the equation written in center-radius form. Writing the equation of a circle given the center and the radius Writing the equation of a circle from standard form to general form and vice-versa

Quiz: Lesson 2 Explaining how to determine the center of a circle Explaining how to graph circles given the equations written in center-radius form and general form Explaining how to write the equation of a circle given the center and the radius Explaining how to write the equation of a circle from standard form to general form and vice-versa Solving problems involving the equation of a circle

Summative Post-Test: Part I Identifying the distance formula Illustrating the distance between two points on the coordinate plane

Post-Test: Part I Determining the distance between a pair of points Determining the coordinate of a point given its distance from another point

Post-Test: Part I and Part II Solving problems involving the Distance Formula, including the Midpoint Formula, and the Equation of a Circle

Post-Test: Part III A and B Preparing emergency measures to be undertaken in times of natural calamities and disasters particularly typhoons and floods

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Illustrating the midpoint formula Illustrating the midpoint of a segment Defining coordinate proof Identifying an equation of a circle

Determining the coordinates of the midpoint and the endpoints of a segment Describing the figure formed by a set of points Determining the coordinates of the vertex of a geometric figure Finding the length of the radius of a circle given the endpoints of a diameter Finding the center of a circle given the equation Finding the equation of a circle given the endpoints of a radius

Preparing a grid map of a municipality Formulating and solving problems involving the key concepts of plane coordinate geometry

Self-Assessment

Journal Writing: Expressing understanding of the distance formula, midpoint formula, coordinate proof, and the equation of a circle.

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Assessment Matrix (Summative Test))

Levels of Assessment What will I assess? How will I

assess? How Will I Score?

Knowledge 15%

The learner demonstrates understanding of key concepts of plane coordinate geometry. Derive the distance

formula. Apply the distance

formula to prove some geometric properties.

Illustrate the center-radius form of the equation of a circle.

Determine the center and radius of a circle given its equation and vice versa.

Graph a circle and other geometric figures on the coordinate plane.

Solve problems involving geometric figures on the coordinate plane.

Paper and Pencil Test Part I items 1, 3, 4, 7, 8, and 13


Process/Skills 25%

Part I items 5, 6, 9, 10, 11, 12, 14, 16, 18, and 19


Understanding 30%

Part I items 2, 15, 17, and 20 Part II items 1 and 2

1 point for every correct response Rubric on Problem Solving (maximum of 4 points for each problem)


30%

The learner is able to formulate and solve problems involving geometric figures on the rectangular coordinate plane with perseverance and accuracy.

Part III A Part III B

Rubric for the Prepared Emergency Measures Rubric for Grip Map of the Municipality (Total Score: maximum of 6 points ) Rubric on Problems Formulated and Solved (Total Score: maximum of 6 points )

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This module covers key concepts of plane coordinate geometry. It is divided into two lessons, namely: The Distance Formula and the Equation of a Circle.

In Lesson 1 of this module, the students will derive the distance formula and apply it in proving geometric relationships and in solving problems, particularly finding the distance between objects or points. They will also learn about the midpoint formula and its applications. Moreover, the students will graph and describe geometric figures on the coordinate plane.

The second lesson is about the equation of a circle. In this lesson, the students will illustrate the center-radius form of the equation of a circle, determine the center and the radius given its equation and vice-versa, and show its graph on the coordinate plane (or by using the computer freeware, GeoGebra). More importantly, the students will solve problems involving the equation of a circle.

In learning the equation of a circle, the students will use their prior knowledge and skills through the different activities provided. This is to connect and relate those mathematics concepts and skills that students previously studied to their new lesson. They will also perform varied learning tasks to process the knowledge and skills learned and to further deepen and transfer their understanding of the different lessons in real-life situations.

Introduce the main lesson to the students by showing them the pictures below, then ask them the questions that follow:

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Entice the students to find the answers to these questions and to

determine the vast applications of plane coordinate geometry through this module. Objectives:

After the learners have gone through the lessons contained in this module, they are expected to: 1. derive the distance formula;

2. find the distance between points;

3. determine the coordinates of the midpoint of a segment;

4. name the missing coordinates of the vertices of some geometric figures;

5. write a coordinate proof to prove some geometric relationships;

6. give/write the center-radius form of the equation of a circle;

7. determine the center and radius of a circle given its equation and vice versa;

8. write the equation of a circle from standard form to general form and vice

versa;

9. graph a circle and other geometric figures on the coordinate plane; and

10. solve problems involving geometric figures on the coordinate plane.

Look around! What geometric figures do you see in your classroom, school buildings, houses, bridges, roads, and other structures? Have you ever asked yourself how geometric figures helped in planning the construction of these structures?

In your community or province, was there any instance when a

stranger or a tourist asked you about the location of a place or a landmark? Were you able to give the right direction and its distance? If not, could you give the right information the next time somebody asks you the same question?

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PRE-ASSESSMENT:

Assess students’ prior knowledge, skills, and understanding of mathematics concepts related to the Distance Formula, the Midpoint Formula, the Coordinate Proof, and the Equation of a Circle. These will facilitate teaching and students’ understanding of the lessons in this module.


Students are expected to demonstrate understanding of key concepts of plane coordinate geometry, formulate real-life problems involving these concepts, and solve these with perseverance and accuracy.

Lesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate Proof

What to KNOW

Check students’ knowledge of the different mathematics concepts previously studied and their skills in performing mathematical operations. These will facilitate teaching and students’ understanding of the distance formula and the midpoint formula and in writing coordinate proofs. Tell them that as they go through this lesson, they have to think of this important question: How do the distance formula, the midpoint formula, and the coordinate proof facilitate finding solutions to real-life problems and making decisions?

Let the students start the lesson by doing Activity 1. Ask them to use the

given number line in determining the lengths of segments. Let them explain how

Answer Key Part I Part II (Use the rubric to rate students’ works/outputs)

1. C 11. D 1. 100 km 2. C 12. A 2. 994

22 yx

3. B 13. A 4. B 14. B 5. B 15. C Part III (Use the rubric to rate students’

works/outputs) 6. D 16. C 7. B 17. C 8. D 18. B 9. A 19. D

10. C 20. B

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they used the coordinates of points in finding each length. Emphasize in this activity the relationships among the segments based on their lengths, the distance between the endpoints of segments whose coordinates on the number line are known, and the significance of these to the lesson.

Activity 1: How long is this part? Answer Key

1. 4 units 2. 4 units 3. 6 units 4. 2 units 5. 3 units 6. 1 unit

a. Counting the number of units from one point to the other point using the number line or finding the absolute value of the difference of the coordinates of the points

b. Yes. By counting the number of units from one point to the other point using the number line or finding the absolute value of the difference of the coordinates of the points

c. AB BC , AC CE , CD DG , AB EG . The two segments have the same lengths.

d. d.1) AB + BC = AC; d.2) AC + CE = AE e. Yes. The absolute values of the difference of their coordinates are

equal. AD = 410 = 14 DA = 104 = 14 BF = 96 = 15 FB = 69 = 15

Students’ understanding of the relationships among the sides of a right triangle is a prerequisite to the derivation of the Distance Formula. In Activity 2, provide the students opportunity to recall Pythagorean theorem by asking them to find the length of the unknown side of a right triangle. Tell them to explain how they arrived at each length of a side.

Activity 2: Why am I right? Answer Key

1. 5 units 2. 12 units 3. 12 units 4. 132 units 7.21 units

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5. 54 units 8.94 units 6. 632 units 15.87 units

The length of the unknown side of each right triangle is obtained by applying the Pythagorean theorem.

Let students relate their understanding of the Pythagorean theorem to finding the distance between objects or points on the coordinate plane. This would help them understand the derivation of the distance formula.

Ask the students to perform Activity 3. In this activity, they will be presented with a situation involving distances of objects or points on a coordinate plane. If possible, let the students find out how the coordinates of points can be used in finding distances between objects. Activity 3: Let’s Exercise! Answer Key

1. 10 km. By applying the Pythagorean theorem. That is, 222 86 c ; c = 10 km.

2. 3 km. distance from City Hall 4,0 to Plaza 4,3 = 30 = 3

9 km. distance from City Hall 4,0 to Emilio’s house 4,9 = 90 = 9 3. 9 km. distance from Jose’s house 0,0 to Gasoline Station 0,9 = 90 = 9 4. 0,0 – Jose’s house 12,3 – Diego’s house

4,9 – Emilio’s house 4,3 – Plaza 5. 4,0 – City Hall 0,9 – Gasoline Station 6. By finding the absolute value of the difference of the coordinates of the

points corresponding to Emilio’s house and the City Hall and Jose’s house and the Gasoline Station, respectively Distance from Emilio’s house 4,9 to City Hall 4,0 = 90

= 9 Answer: 9 km

Distance from Jose’s house 0,0 to Gasoline Station 0,9 = 09

= 9 km Answer: 9 km

The distances of the houses of Jose, Emilio, and Diego from each other can be determined by applying the Pythagorean Theorem.

Jose’s house 0,0 to Emilio’s house 4,9

210

222 94 c ; c = 97 km 9.85 km Jose’s house 0,0 to Diego’s house 12,3

222 123 c ; c = 153 km 12.37 km Emilio’s house 4,9 to Diego’s house 12,3

222 86 c ; c = 10 km

Provide the students opportunity to derive the Distance Formula. Ask them to perform Activity 4. In this activity, the students should be able to come up with the Distance Formula starting from two given points on the coordinate plane. Activity 4: Let Me Formulate!

Answer Key 1. 2.

3. C 1,8 . By determining the coordinates of the point of intersection of the two lines AC = 6 units BC = 8 units

4. Right Triangle. ACBC . Hence,

the triangle contains a 90-degree angle. Pythagorean Theorem can be applied. AB = 10 units

x

y

x

y

x

y

211

Before proceeding to the next activities, let the students give a brief

summary of the activities done. Provide them an opportunity to relate or connect their responses in the activities given to their new lesson. Let the students read and understand some important notes on the distance formula and the midpoint formula and in writing coordinate proofs. Tell them to study carefully the examples given. What to PROCESS

In this section, let the students apply the key concepts of the Distance Formula, Midpoint Formula, and Coordinate Proof. Tell them to use the mathematical ideas and the examples presented in the preceding section to answer the activities provided.

Ask the students to perform Activity 5. In this activity, the students will

determine the distance between two points on the coordinate plane using the Distance Formula. They should be able to explain how to find the distance between points that are aligned horizontally, vertically, or neither.

5. C 21,yx

AC = 21 xx or 12 xx BC = 21 yy or 12 yy

2AB = 212

212 yyxx

AB = 2122

12 yyxx

x

y

212

Activity 5: How far are we from each other?

Let the students apply the Midpoint Formula in finding the coordinates of

the midpoint of a segment whose endpoints are given by doing Activity 6. This activity will enhance their skill in proving geometric relationships using coordinate proof and in solving real-life problems involving the midpoint formula. Activity 6: Meet Me Halfway!

Answer Key 1. 9,9 6. 9,8 2. 8,7 7. 4,5

3. 4,4 8.

215,

215

4. 1,4 9. 7,8

5.

25,

23 10. 4,5

Answer Key

1. 8 units 6. 13 units 2. 15 units 7. 10.3 units 3. 11.4 units 8. 66.11 units 4. 13 units 9. 13.6 units 5. 6.4 units 10. 12.81 units

a. Regardless of whether points are aligned horizontally or vertically,

the distance d between these points can be determined using the

Distance Formula, 2122

12 yyxxd . Moreover, the following formulas can also be used.

a.1) d = 12 xx , for the distance d between two points that are aligned horizontally a.2) d = 12 yy , for the distance d between two points that are aligned vertically

b. The Distance Formula can be used to find the distance between two points on a coordinate plane.

213

Provide the students opportunity to relate the properties of some geometric figures to the new lesson by performing Activity 7. Ask them to plot some set of points on the coordinate plane. Then, connect the consecutive points by a line segment to form a figure. Tell them to identify the figures formed and use the distance formula to characterize or describe each. Emphasize to the students the different properties of these geometric figures for they need this in determining the missing coordinates of each figure’s vertices.

Activity 7: What figure am I?

Answer Key 1. 2. 3. 4. y

x

y

x

x

y

x

y

214

5. 6.

x

y

x

y

7. 8.

x

y

x

y

9. 10.

x

y

x

y

215

An important skill that students need in writing coordinate proof is to name

the missing coordinates of geometric figures drawn on a coordinate plane. Activity 8 provides the students opportunity to develop such skill. In this activity, the students will name the missing coordinates of the vertices of geometric figures in terms of the given variables.

Activity 8: I Missed You But Now I Found You!


Answer Key 1. O cba , 5. A 0,a 2. V ba, D da, 3. V 0,3a E cb,

M ba,3 6. S 0,0 4. W cb, P ba,

For questions a-d, evaluate students’ responses.

a. The figures formed in #1, #2, and #3 are triangles. Each figure has three sides. The figures formed in #4, #5, #6, #7, #8, and #9 are quadrilaterals. Each figure has four sides. The figure formed in #10 is a pentagon. It has five sides.

b. ΔABC and ΔFUN are isosceles triangles. ΔGOT and ΔFUN are right triangles.

c. ΔABC and ΔFUN are isosceles because each has two sides congruent or with equal lengths. ΔGOT and ΔFUN are right triangles because each contains a right angle.

d. Quadrilaterals LIKE and LOVE are squares. Quadrilaterals LIKE, DATE, LOVE and SONG are rectangles. Quadrilaterals LIKE, DATE, LOVE, SONG, and BEAT are parallelograms. Quadrilateral WIND is a trapezoid.

e. Quadrilaterals LIKE and LOVE are squares because each has four sides congruent and contains four right angles. Quadrilaterals LIKE, DATE, LOVE, and SONG are rectangles because each has two pairs of congruent and parallel sides and contains four right angles. Quadrilaterals LIKE, DATE, LOVE, SONG, and BEAT are parallelograms because each has two pairs of congruent and parallel sides and has opposite angles that are congruent. Quadrilateral WIND is a trapezoid because it has a pair of parallel sides.

216

Ask the students to take a closer look at some aspects of the Distance

Formula, the Midpoint Formula, and the Coordinate Proof. Provide them with opportunities to think deeply and test further their understanding of the lesson by doing Activity 9. In this activity, the students will solve problems involving these mathematics concepts and explain or justify their answers.

Activity 9: Think of This Over and Over and Over … Again! Answer Key 1. y = 15 or y = -9; 2. a. x = 21 – if N is in the first quadrant

x = -3 – if N is in the second quadrant

b.

25,3

3. 4,7 4. 99 km 5. Luisa and Grace are both correct. If the expressions are evaluated,

Luisa and Grace will arrive at the same value. 6. a. Possible answer: To become more accessible to students coming

from both buildings. b. 70,90 c. The distance between the two buildings is about 357.8 m.

Since the study shed is midway between the two school buildings, then it is about 178.9 m away from each. This is obtained by dividing 357.8 by 2.

7. a. 100 km b. 5 hours 8. No. The triangle is not an equilateral triangle. It is actually an isosceles

triangle. The distance between A and C is 2a while the distance between A and B or B and C is 2a .

9. a. Yes. 22dbacFS and 22

dbcaAT .

Since 22acca , then FS = AT.

b. Rectangle; The quadrilateral has two pairs of opposite sides that are parallel and congruent and has four right angles.

Develop further students’ understanding of Coordinate Proof by asking

them to perform Activity 10. Ask the students to write a coordinate proof to prove the particular geometric relationship. Let them realize the significance of the Distance Formula, the Midpoint Formula, and the different mathematics concepts already studied in coming up with the coordinate proof.

The values of x were obtained by using the distance formula and the coordinates of the midpoint were determined by using the midpoint formula. Students may further give explanations to their answers based on the solutions presented.

217

Activity 10: Prove that this is True! Answer Key 1. Show that QSPR .

If QSPR , then QSPR .

202 cabPR

2222 caabb

2222 cbabaPR

22 0 cabQS

22 0 cab

2222 caabb

2222 cbabaQS

Therefore, QSPR and QSPR . Hence, the diagonals of an isosceles trapezoid are congruent.

2. Show that LGMC2

1 .

2

02

20

2

baMC

2

4

2

4

ba

2

22 baMC

2020 baLG

22 ba

2

22

2

1 baLG

Therefore, LGMC2

1 . Hence, the median to the hypotenuse of a right

triangle is half the hypotenuse.

218

3. Show that PSRSQRPQ .

2

2

2

20

cc

abPQ

2

2

2

2

cab

2

2222 caabbPQ

2

02

20

2

cabQR

2

20

2

20

cabRS

2

2

2

2

cab 2

2

2

2

cab

2

2222 caabbQR

2

2222 caabbRS

2

2

2

20

cc

abPS

2

2

2

2

cab

2

2222 caabbPS

Therefore, PSRSQRPQ and PQRS is a rhombus.

219

4. Show that CSBT . If CSBT , then CSBT .

22

20

2

baaBT

22

20

2

baa

2

2

2

2

3

ba

2

229 baBT

2

20

2

2

baaCS

2

2

2

2

3

ba

2

229 baCS

5. Equate the lengths AC and BD to prove that ABCD is a rectangle.

BDAC

2020202 cbacab

22222222 cbabacaabb

22222222 cbabacaabb abab 22

04 ab Since a > 0, then b = 0. And that A is along the y – axis. Also, B is along the line parallel to the y-axis. Therefore, ADC is a right angle and ABCD is a rectangle.

Therefore, CSBT and CSBT . Hence, the medians to the legs of an isosceles triangle are congruent.

220

6. Show that LECG2

1

2020 cbLE

22 cbLE

2

02

2

22

cabaCG

2

2

2

2

cb

2

22 cbCG

Therefore, LECG2

1 .

Before the students move to the next section of this lesson, give a short

test (formative test) to find out how well they understood the lesson. Ask them also to write a journal about their understanding of the distance formula, midpoint formula, and the coordinate proof. Refer to the Assessment Map. What to TRANSFER

Give the students opportunities to demonstrate their understanding of the Distance Formula, the Midpoint Formula, and the use of Coordinate Proofs by doing a practical task. Let them perform Activity 11. You can ask the students to work individually or in group. In this activity, the students will make a sketch of the map of their municipality, city, or province on a coordinate plane. They will indicate on the map some important landmarks, and then determine the coordinates of each. Tell them to explain why the landmarks they have indicated are significant in their community and to write a paragraph explaining how they selected the coordinates of these landmarks. Using the coordinates assigned to the different landmarks, the students will formulate then solve problems involving the distance formula and the midpoint formula. They will also formulate problems which require the use of coordinate proofs. Activity 11: A Map of My Own

Answer Key

Evaluate students’ answers. You may use the rubric.

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Summary/Synthesis/Generalization: This lesson was about the distance formula, the midpoint formula, the use

of coordinate proofs, and the applications of these mathematical concepts in real life. The lesson provided the students with opportunities to derive the distance formula, find the distance between points, determine the coordinates of the midpoint of a segment, name the missing coordinates of the vertices of some geometric figures, write a coordinate proof to prove some geometric relationships, and solve problems involving the different concepts learned in this lesson. Moreover, the students were given the opportunities to formulate then solve problems involving the distance formula, the midpoint formula, and the coordinate proof. Lesson 2: The Equation of a Circle

What to KNOW

Find out how much the students have learned about the different mathematics concepts previously studied and their skills in performing mathematical operations. Checking these will facilitate teaching and students’ understanding of the equation of a circle. Tell them that as they go through this lesson, they have to think of this important question: “How does the equation of a circle facilitate finding solutions to real-life problems and making decisions?”

Two of the essential mathematics concepts needed by the students in

understanding the equation of a circle are the perfect square trinomial and the square of a binomial. Activity 1 of this lesson will provide them opportunity to recall these concepts. In this activity, the students will determine the number that must be added to a given expression to make it a perfect square trinomial and then express the result as a square of a binomial. They should be able to explain how they came up with the perfect square trinomial and the square of a binomial. Emphasize to the students that the process they have done in producing a perfect square trinomial is also referred to as completing the square. Activity 1: Make It Perfect!

Answer Key 1. 4; 22x

2. 25; 25t

3. 49; 27r

4. 121; 211r

5. 324; 218x

a. Add the square of one-half the coefficient of the linear term.

b. Factor the perfect square trinomial. c. Use the distributive property of

multiplication or FOIL Method.

222

Provide the students opportunity to develop their understanding of the equation of a circle. Ask them to perform Activity 2. In this activity, the students will be presented with a situation involving the equation of a circle. Let them find the distance of the plane from the air traffic controller given the coordinates of the point where it is located and the y-coordinate of the position of the plane at a particular instance if its x-coordinate is given. Furthermore, ask them to describe the path of the plane as it goes around the airport. Challenge them to determine the equation that would define the path of the plane. Let them realize that the distance formula is related to the equation defining the plane’s path around the airport.

Activity 2: Is there a traffic in the air? Answers Key

Provide the students opportunity to come up with an equation that can be

used in finding the radius of a circle. Ask them to perform Activity 3. In this activity, the students should be able to realize that the Distance Formula can be used in finding the radius of a circle. And that the distance of a point from the center of a circle is also the radius of the circle.

Answer Key

1. 50 km 2. When x = 5, y = 49.75 or y = -49.75.

When x = 10, y = 48.99 or y = -48.99. When x = 15, y = 47.7 or y = -47.4 When x = 15, y = 47.7 or y = -47.4 When x = -20, y = 45.83 or y = -45.83. When x = -30, y = 40 or y = -40.

3. No. It is not possible for the plane to be at a point whose coordinatex is 60 because its distance from the air traffic controller would be greater than 50 km.

4. The path is circular. 250022 yx

Answer Key

6. 481 ;

2

29

w 9.

361 ;

2

61

s

7. 4

121; 2

211

x 10.

649 ;

2

83

t

8. 4

625 ; 2

225

v

223

Activity 3: How far am I from my point of rotation? A.

Answer Key

1. 8 units 2. Yes, the circle will pass through

8,0 , 0,8 , and 8,0 because the distance from these points to the center of the circle is 8 units.

3. No, because the distance from point M 6,4 to the center of the circle is less than 8 units. No, because the distance from point N 2,9 to the center of the circle is more than 8 units.

4. 8 units; 08 = 8 5. If a point is on the circle, its distance from the center is equal to the

radius. 6. Since the distance d of a point from the center of the circle is

22 yxd and is equal to the radius r, then 22 yxr or 222 ryx .

y

x

224

B.

Before proceeding to the next activities, let the students give a brief summary of the activities they have done. Provide them with an opportunity to relate or connect their responses in the activities given to their new lesson, equation of a circle. Let the students read and understand some important notes on equation of a circle. Tell them to study carefully the examples given. What to PROCESS

Let the students use the mathematical ideas they have learned about the equation of a circle and the examples presented in the preceding section to perform the succeeding activities.

Answer Key

1. 61 units or approximately 7.81 units

2. Yes, the circle will pass through 7,2 , 7,8 , and 4,3 because the distance from each of these points to the center of the circle is 61 units or approximately 7.81 units.

3. No, because the distance from point M 6,7 to the center of the circle is more than 7.81 units.

4. 61 units or approximately 7.81 units. Note: Evaluate students’ explanations.

5. If the center of the circle is not at the origin, its radius can be determined by using the distance formula,

2122

12 yyxxd . Since the distance of the point from the center of the circle is equal to the radius r, then

2122

12 yyxxr or 2212

212 ryyxx . If

y,xP is a point on the circle and k,hC is the center, then

2212

212 ryyxx becomes 222 rkyhx .

y

x

225

In Activity 4, the students will determine the center and the radius of each circle, given its equation. Then, the students will be asked to graph the circle. Ask them to explain how they determined the center and the radius of the circle. Furthermore, tell them to explain how to graph a circle given its equation in different forms. Strengthen students’ understanding of the graphs of circles through the use of available mathematics freeware like Geogebra. Activity 4: Always Start at This Point!

Answer Key

1. Center: 0,0 3. Center: 0,0 Radius: 7 units Radius: 10 units


y

x

y

x

y

x

y

x

226

Ask the students to perform Activity 5. This time, the students will write the

equation of a circle given the center and the radius. Ask them to explain how to determine the equation of a circle whether or not the center is the origin. Activity 5: What defines me? Answer Key 1. 14422 yx

2. 8162 22 yx

3. 22527 22 yx

4. 5054 22 yx

5. 27810 22 yx

Answer Key


a. Note: Evaluate students’ responses. b. Determine first the center and the radius of the circle defined by the

equation, then graph. If the given equation is in the form 222 ryx , the center is at the origin and the radius of the circle is r. If the given equation is in the form 222 rkyhx , the center is at kh, and the radius of the circle is r.

If the given equation is in the form 022 FEyDxyx ,

transform it into the form 222 rkyhx . The center is at kh, and the radius of the circle is r.

a. Write the equation in the

form 222 ryx where the origin is the center and r is the radius of the circle. Write the equation in the

form 222 rkyhx where kh, is the center and r is the radius of the circle.

b. No, because the two circles have different radii.

y

x

y

x

227

Activities 6 and 7 provide students opportunities to write equations of circles from center-radius form or standard form to general form and vice-versa. At this point, ask them to explain how to transform the equation of a circle from one form to another form and discuss the mathematics concepts or principles applied. Furthermore, challenge them to find a shorter way of transforming equation of a circle from general form to standard form and vice-versa. Activity 6: Turn Me into a General!

Answer Key 1. 0168422 yxyx 6. 0151422 xyx

2. 04718822 yxyx 7. 045422 yyx

3. 04421222 yxyx 8. 096422 xyx

4. 0112141622 yxyx 9. 023101022 yxyx

5. 0111022 yyx 10. 08822 yxyx Note: Evaluate students’ explanations.

Activity 7: Don’t Treat this as a Demotion!

Answer Key 1. 6441 22

yx 4. 1004 22 yx Center: 4,1 Center: 4,0 Radius: 8 units Radius: 10 units

2. 3622 22 yx 5. 4

31

32 22

yx

Center: 2,2 Center:

31,

32

Radius: 6 units Radius: 2 units

3. 3225 22 yx 6. 9

23

25 22

yx

Center: 2,5 Center:

23,

25

Radius: 24 units Radius: 3 units a. Grouping the terms, then applying completing the square, addition

property of equality and factoring.

228

b. Completing the square, Addition Property of Equality, Square of a Binomial

c. Using the values of D, E, and F in the general equation of a circle,

022 FEyDxyx , to find the center (h,k) and radius r. The GeoGebra freeware can also be used for verification.

What to REFLECT on and UNDERSTAND:

Ask the students to have a closer look at some aspects of the equation of

a circle. Provide them with opportunities to think deeply and test further their understanding of the equation of a circle by doing Activities 8 and 9. Give more focus on the real-life applications of the equation of a circle. Activity 8: A Circle? Why not?

Activity 9: Find Out More!

Answer Key 1. No. 0268222 yxyx can be written as 92421 yx .

Notice that -9 cannot be expressed as a square of another number.

2. Yes. yxyx 104922 can be written as 202522 yx .

3. No. 328622 yxyx is not an equation of a circle. Its graph is not also a circle.

4. No. 06514822 yxyx is merely a point. The radius must be greater than 0 for a circle to exist.

Answer Key 1. 8183 22

yx

2. 36710 22 yx or 36510 22

yx 3. 753 yx

4. 1355 22 yx

5. a. 10043 22 yx

b. Yes, because point 6,11 is still within the critical area. c. Follow the advice of PDRRMC. d. (Evaluate students’ responses/explanations.)

229

Before the students move to the next section of this lesson, give a short

test (formative test) to find out how well they understood the lesson. Ask them also to write a journal about their understanding of the equation of a circle. Refer to the Assessment Map.

What to TRANSFER

Give the students opportunities to demonstrate their understanding of the equation of a circle by doing a practical task. Let them perform Activity 10. You can ask the students to work individually or in a group.

In Activity 10, the students will paste some small pictures of objects on

grid paper and position them at different coordinates. Then, the students will draw circles that contain these pictures. Using the pictures and the circles drawn on the grid, they will formulate problems involving the equation of the circle, and then solve them.

Activity 10: Let This be a Part of My Scrapbook!

Answer Key Evaluate students’ answers. You may use the rubric.

Answer Key

6. a. Wise Tower - 8135 22 yx

Global Tower - 1663 22 yx

Star Tower - 36312 22 yx

b. 2,12 - Star Tower 7,6 - Wise Tower 8,2 - Global Tower 3,1 - Wise and Global Tower

c. Many possible answers. Evaluate students’ responses.

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This lesson was about the equation of circles. The lesson provided the

students with opportunities to illustrate the center-radius form of the equation of a circle, determine the center and the radius of a circle given its equation and vice versa, write the equation of a circle from standard form to general form and vice-versa, graph circles on the coordinate plane, and solve problems involving the equation of circles. Moreover, they were given the opportunity to formulate and solve real-life problems involving the equation of a circle through the practical task performed. Their understanding of this lesson and other previously learned mathematics concepts and principles will facilitate their learning of other related mathematics concepts.

231

SUMMATIVE TEST

Part I Choose the letter that you think best answers the question. 1. Which of the following is NOT a formula for finding the distance between two

points on the coordinate plane?

A. 12 xxd C. 2122

12 yyxxd

B. 12 yyd D. 2122

12 yyxxd

2. A map is drawn on a grid where 1 unit is equivalent to 2 km. On the same map, the coordinates of the point corresponding to San Rafael is (1,4). Suppose San Quintin is 20 km away from San Rafael. Which of the following could be the coordinates of the point corresponding to San Rafael? A. (17,16) B. (17,10) C. (9,10) D. (-15,16)

3. Let M and N be points on the coordinate plane as shown in the figure below.

If the coordinates of M and N are 75, and 45 , , which of the following would give the distance between the two points? A. 47 B. 57 C. 74 D. 54

4. Point Q is the midpoint of ST . Which of the following is true about ST?

A. QTQSST C. QTQSST 2 B. QTQSST D. QTQSST 2

5. The distance between points 5,xM and 15 ,C is 10 units. What is the x-

coordinate of M if it lies in the second quadrant? A. -7 B. -3 C. -1 D. 13

x

y

232

6. What is the distance between points D(-10,2) and E(6,10)? A. 16 B. 20 C. 210 D. 58

7. Which of the following equation describes a circle on the coordinate plane with a center at 32 , and a radius of 5 units?

A. 2222532 yx C. 222

2523 yx

B. 222532 yx D. 222

532 yx

8. Which of the following would give the coordinates of the midpoint of P(-6,13) and Q(9,6)?

A.

2

69

2

136, C.

2

69

2

136,

B.

2

613

2

96, D.

2

613

2

96,

9. The endpoints of a segment are (-5,2) and (9,12), respectively. What are the

coordinates of its midpoint? A. (7,5) B. (2,7) C. (-7,5) D. (7,2)

10. The coordinates of the vertices of a rectangle are 62,W , 610,I ,

310 ,N , and 32 ,D . What is the length of a diagonal of the rectangle? A. 7.5 B. 9 C. 12 D. 15

11. The coordinates of the vertices of a triangle are 24,G , 15 ,O , and 810,T . What is the length of the segment joining the midpoint of GT and

O? A. 102 B. 58 C. 103 D. 106

12. The endpoints of a diameter of a circle are 86,E and 24 ,G . What is the length of the radius of the circle? A. 210 B. 25 C. 102 D. 10

13. What proof uses figures on a coordinate plane to prove geometric properties? A. Indirect Proof C. Coordinate Proof B. Direct Proof D. Two-Column Proof

14. What figure is formed when the points K(-2,10), L(8,8), M(6,2), and N(-4,4)

are connected consecutively? A. Trapezoid B. Parallelogram C. Square D. Rectangle

233

15. Three speed cameras were installed at different points along an expressway. On a map drawn on a coordinate plane, the coordinates of the first speed camera are (-2,4). Suppose the second camera is exactly between the other two and its coordinates are (12,8). What are the coordinates of the third speed camera?

A. (26,12) B. (26,16) C. (22,12) D. (22,16)

16. In the equilateral triangle below, what are the coordinates of P? A. a,20 B. 02 ,a C. 30 a, D. 20 a,

17. Jose, Andres, Emilio, and Juan live in different barangays of Magiting town as

shown on the coordinate plane below. Who lives the farthest from the Town Hall if it is located at the origin? A. Jose B. Andres C. Emilio D. Juan

Jose

Emilio

Andres

Juan

Town Hall

234

18. What is the center of the circle 0366422 yxyx ? A. (9,-3) B. (3,-2) C. (2,-3) D. (2,-10)

19. A radius of a circle has endpoints 34, and 21, . What is the equation

that defines the circle if its center is at the second quadrant? A. 5021

22 yx C. 5034

22 yx

B. 502122 yx D. 5034

22 yx

20. A radio signal can transmit messages up to a distance of 5 km. If the radio

signal’s origin is located at a point whose coordinates are (-2,7). What is the equation of the circle that defines the boundary up to which the messages can be transmitted? A. 2572

22 yx C. 2572

22 yx

B. 57222 yx D. 572

22 yx

Part II Directions: Solve each of the following problems. Show your complete solutions. 1. A tracking device that is installed in a mobile phone indicates that its user is

located at a point whose coordinates are (18,14). In the tracking device, each unit on the grid is equivalent to 7 km. If the phone user came from a place whose coordinates are (2,6)? How far has he travelled?

2. The equation that represents the transmission boundaries of a cellular phone

tower is 019921022 yxyx . What is the greatest distance, in kilometers, can the signal of the tower be transmitted?

Rubric for Problem Solving

4 3 2 1 Used an appropriate strategy to come up with correct solution and arrived at a correct answer

Used an appropriate strategy to come up with a solution, but a part of the solution led to an incorrect answer

Used an appropriate strategy but came up with an entirely wrong solution that led to an incorrect answer

Attempted to solve the problem but used an inappropriate strategy that led to a wrong solution

235

Part III A: GRASPS Assessment Perform the following. Goal: To prepare emergency measures to be undertaken in times of

natural calamities and disasters particularly typhoons and floods Role: Radio Group Chairman of the Municipal Disaster and Risk

Management Committee Audience: Municipal and Barangay Officials and Volunteers Situation: Typhoons and floods frequently affect your municipality during

rainy seasons. For the past years, losses of lives and damages to properties have occurred. Because of this, your municipal mayor designated you to chair the Radio Group of the Municipal Disaster and Risk Management Committee to warn the residents of your municipality of any imminent natural calamities and disasters like typhoons and floods. The municipal government gave your group a number of two-way radios and antennas to be installed in strategic places in the municipality. These shall be used as the need arises.

As chairman of the Radio Group, you were tasked to prepare

emergency measures that you will undertake to reduce if not to avoid losses of lives and damages to properties during rainy seasons. These include the positioning of the different two-way radios and antennas for communication and coordination among the members of the Radio Group. You were also asked to prepare a grid map of your municipality showing the positions of the two-way radios and antennas.

Products: 1. Emergency Measures to be undertaken in times of natural calamities and disasters

2. Grid map of your municipality showing the locations of the

different two-way radios and antennas Standards: The emergency measures must be clear, relevant, and systematic.

The grid map of the municipality must be accurate, presentable, and appropriate.

236

Rubric for the Prepared Emergency Measures

4 3 2 1 The emergency measures are clearly presented, relevant to the situation, and systematic.

The emergency measures are clearly presented and relevant to the situation but not systematic.

The emergency measures are clearly presented but not relevant to the situation and not systematic.

The emergency measures are not clearly presented, not relevant to the situation, and not systematic.

Rubric for Grid Map of the Municipality

4 3 2 1 The grid map is accurately made, appropriate, and presentable.

The grid map is accurately made and appropriate but not presentable.

The grid map is not accurately made but appropriate.

The grid map is not accurately made and not appropriate.

Part III B Use the prepared grid map of the municipality in Part III A in formulating problems involving plane coordinate geometry, then solve. Rubric on Problems Formulated and Solved

Score Descriptors

6 Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate.

5

Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

4

Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.

3

Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or

237

Score Descriptors misinterprets less significant ideas or details.

2

Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension.

1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach.

Source: D.O. #73, s. 2012

Glossary of Terms Coordinate Proof – a proof that uses figures on a coordinate plane to prove geometric relationships. Distance Formula – an equation that can be used to find the distance between any pair of points on the coordinate plane. The distance formula is

2122

12 yyxxd or 2122

12 yyxxPQ , if 11 y,xP and 22 y,xQ are points on a coordinate plane.

Horizontal Distance (between two points) – the absolute value of the difference of the x-coordinates of two points

Midpoint – a point on a line segment that divides the same segment into two equal parts. Midpoint Formula – a formula that can be used to find the coordinates of the midpoint of a line segment on the coordinate plane. The midpoint of 11 y,xP

and 22 y,xQ is

222121 yy,xx

.

Answer Key Part I Part II (Use the rubric to rate students’ works/outputs) 1. C 11. A 1. 556 km 2. C 12. B 2. 15 km 3. C 13. C 4. A 14. B 5. B 15. A Part III A (Use the rubric to rate students’ works/outputs) 6. D 16. C Part III B (Use the rubric to rate students’ works/outputs) 7. D 17. C 8. B 18. C 9. B 19. C

]10. D 20. C

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The General Equation of a Circle – the equation of a circle obtained by expanding 222

rkyhx . The general equation of a circle is

022 FEyDxyx , where D, E, and F are real numbers. The Standard Equation of a Circle – the equation that defines a circle with center at (h, k) and a radius of r units. It is given by

2 2 2. x h y k r Vertical Distance (between two points) – the absolute value of the difference of the y-coordinates of two points. DepEd INSTRUCTIONAL MATERIALS THAT CAN BE USED AS ADDITIONAL RESOURCES: 1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third

Year Mathematics. Plane Coordinate Geometry. Module 20: Distance and Midpoint Formulae

2. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third

Year Mathematics. Plane Coordinate Geometry. Module 22: Equation of a Circle

3. Distance Learning Module (DLM) 3, Module 3: Plane Coordinate Geometry.

4. EASE Modules Year III, Module 2: Plane Coordinate Geometry References and Website Links Used in This Module: References: Bass, L. E., Charles, R. I., Hall, B., Johnson, A., & Kennedy, D. (2008) Texas

Geometry. Boston, Massachusetts: Pearson Prentice Hall. Bass, L. E., Hall, B.R., Johnson, A., & Wood, D. F. (1998) Prentice Hall

Geometry Tools for a Changing World. NJ, USA: Prentice-Hall, Inc. Boyd, C., Malloy, C. & Flores. (2008) Glencoe McGraw-Hill Geometry. USA: The

McGraw-Hill Companies, Inc. Callanta, M. M. (2012) Infinity, Worktext in Mathematics III. Makati City: EUREKA

Scholastic Publishing, Inc.

239

Chapin, I., Landau, M. & McCracken. (1997) Prentice Hall Middle Grades Math, Tools for Success. Upper Saddle River, New Jersey: Prentice-Hall, Inc.

Cifarelli, V. (2009) cK-12 Geometry, Flexbook Next Generation Textbooks. USA:

Creative Commons Attribution-Share Alike. Clemens, S. R., O’Daffer, P. G., Cooney, T. J., & Dossey, J. A. (1990) Addison-

Wesley Geometry. USA: Addison-Wesley Publishing Company, Inc. Clements, D. H., Jones, K. W., Moseley, L.G., & Schulman, L. (1999) Math in my

World. New York: McGraw-Hill Division. Department of Education. (2012) K to 12 Curriculum Guide Mathematics.

Philippines. Gantert, A. X. (2008) AMSCO’s Geometry. NY, USA: AMSCO School

Publications, Inc. Renfro, F. L. (1992) Addison-Wesley Geometry Teacher’s Edition. USA:

Addison-Wesley Publishing Company, Inc. Rich, B. & Thomas, C. (2009) Schaum’s Outlines Geometry Fourth Edition. USA:

The McGraw-Hill Companies, Inc. Smith, S. A., Nelson, C.W., Koss, R. K., Keedy, M. L., & Bittinger, M. L. (1992)

Addison-Wesley Informal Geometry. USA: Addison-Wesley Publishing Company, Inc.

Wilson, P. S. (1993) Mathematics, Applications and Connections, Course I.

Westerville, Ohio: Glencoe Division of Macmillan/McGraw-Hill Publishing Company.

240

Website Links as References and Sources of Learning Activities: CliffsNotes. Midpoint Formula. (2013). Retrieved from http://www.cliffsnotes.com/math/geometry/coordinate-geometry/midpoint-formula CliffsNotes. Distance Formula. (2013). Retrieved from http://www.cliffsnotes.com/math/geometry/coordinate-geometry/distance-formula Math Open Reference. Basic Equation of a Circle (Center at 0,0). (2009). Retrieved from http://www.mathopenref.com/ coordbasiccircle.html Math Open Reference. Equation of a Circle, General Form (Center anywhere). (2009). Retrieved from http://www.mathopenref.com/coordgeneralcircle.html Math-worksheet.org. Using equations of circles. (2014). Retrieved from http://www.math-worksheet.org/using-equations-of-circles Math-worksheet.org. Writing equations of circles. (2014). Retrieved from http://www.math-worksheet.org/writing-equations-of-circles Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved from http://www.regentsprep.org/Regents/ math/geometry/GCG2/ Lmidpoint.htm Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved from http://www.regentsprep.org/Regents/math/geometry/GCG3/ Ldistance.htm Stapel, Elizabeth. "Conics: Circles: Introduction & Drawing." Purplemath. Retrieved from http://www.purplemath.com/modules/ circle.htm Website Links for Videos: Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem Khan Academy. Completing the square to write equation in standard form of a circle. Retrieved from https://www.khanacademy.org/math/ geometry/cc-geometry-circles/equation-of-a-circle/v/completing-the-square-to-write-equation-in-standard-form-of-a-circle Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem

http://www.mathopenref.com/

http://www.regentsprep.org/Regents/

http://www.regentsprep.org/Regents/math/geometry/GCG3/

http://www.purplemath.com/modules/

https://www.khanacademy.org/math/geometry/

https://www.khanacademy.org/math/


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Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theorem Ukmathsteacher. Core 1 – Coordinate Geometry (3) – Midpoint and distance formula and Length of Line Segment. Retrieved from http://www.youtube.com/watch?v=qTliFzj4wuc VividMaths.com. Distance Formula. Retrieved from http://www.youtube.com/watch?v=QPIWrQyeuYw Website Links for Images: asiatravel.com. Pangasinan Map. Retrieved from http://www.asiatravel.com/philippines/pangasinan/pangasinanmap.jpg DownTheRoad.org. Pictures of, Chengdu to Kangding, China Photo, Images, Picture from. (2005). Retrieved from http://www.downtheroad.org/Asia/Photo/ 9Sichuan_China_Image/3Chengdu_Kangding_China.htm funcheap.com. globe-map-wallpapers_5921_1600[1]. Retrieved from http://sf.funcheap.com/hostelling-internationals-world-travel-101-santa-clara/globe-map-wallpapers_5921_16001/ Hugh Odom Vertical Consultants. eleven40 theme on Genesis Framework· WordPress. Cell Tower Development – How Are Cell Tower Locations Selected? Retrieved from http://blog.thebrokerlist.com/cell-tower-development-how-are-cell-tower-locations-selected/ LiveViewGPS, Inc. GPS Tracking PT-10 Series. (2014). Retrieved from http://www.liveviewgps.com/gps+tracking+device+pt-10+series.html Sloan, Chris. Current "1991" Air Traffic Control Tower at Amsterdam Schiphol Airport – 2012. (2012). Retrieved from http://airchive.com/html/airplanes-and-airports/amsterdam-schipol-airport-the-netherlands-/current-1991-air-traffic-control-tower-at-amsterdam-schiphol-airport-2012-/25510 wordfromthewell.com. Your Mind is Like an Airplane. (2012). Retrieved from http://wordfromthewell.com/2012/11/14/your-mind-is-like-an-airplane/


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