chapter 7 resource masters - math problem...

Chapter 7Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 7 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860184-3 GeometryChapter 7 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix

Lesson 7-1Study Guide and Intervention . . . . . . . . 351–352Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 353Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 354Reading to Learn Mathematics . . . . . . . . . . 355Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 356







Chapter 7 AssessmentChapter 7 Test, Form 1 . . . . . . . . . . . . 393–394Chapter 7 Test, Form 2A . . . . . . . . . . . 395–396Chapter 7 Test, Form 2B . . . . . . . . . . . 397–398Chapter 7 Test, Form 2C . . . . . . . . . . . 399–400Chapter 7 Test, Form 2D . . . . . . . . . . . 401–402Chapter 7 Test, Form 3 . . . . . . . . . . . . 403–404Chapter 7 Open-Ended Assessment . . . . . . 405Chapter 7 Vocabulary Test/Review . . . . . . . 406Chapter 7 Quizzes 1 & 2 . . . . . . . . . . . . . . . 407Chapter 7 Quizzes 3 & 4 . . . . . . . . . . . . . . . 408Chapter 7 Mid-Chapter Test . . . . . . . . . . . . 409Chapter 7 Cumulative Review . . . . . . . . . . . 410Chapter 7 Standardized Test Practice . 411–412Unit 2 Test/Review (Ch. 4–7) . . . . . . . . 413–414First Semester Test (Ch. 1–7) . . . . . . . 415–416

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A34

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 7 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 7 Resource Masters includes the core materials neededfor Chapter 7. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.

WHEN TO USE Give these pages tostudents before beginning Lesson 7-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

© Glencoe/McGraw-Hill v Glencoe Geometry

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

Assessment OptionsThe assessment masters in the Chapter 7Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 398–399. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

77

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

ambiguous case

angle of depression

angle of elevation

cosine

geometric mean

Law of Cosines

Law of Sines

Pythagorean identity

puh·thag·uh·REE·ahn

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

Pythagorean triple

reciprocal identity

ri·SIP·ruh·kuhl

sine

solve a triangle

tangent

trigonometric identity

trig·uh·nuh·MET·rik

trigonometric ratio

trigonometry

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

77

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

77

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 7. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 7.1

Theorem 7.2

Theorem 7.3

Theorem 7.4Pythagorean Theorem

Theorem 7.5Converse of the Pythagorean Theorem

Theorem 7.6

Theorem 7.7

Study Guide and InterventionGeometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

© Glencoe/McGraw-Hill 351 Glencoe Geometry

Less

on

7-1

Geometric Mean The geometric mean between two numbers is the square root oftheir product. For two positive numbers a and b, the geometric mean of a and b is the positive number x in the proportion �

ax� � �b

x�. Cross multiplying gives x2 � ab, so x � �ab�.

Find the geometric mean between each pair of numbers.

a. 12 and 3Let x represent the geometric mean.

�1x2� � �3

x� Definition of geometric mean

x2 � 36 Cross multiply.

x � �36� or 6 Take the square root of each side.

b. 8 and 4Let x represent the geometric mean.

�8x� � �4

x�

x2 � 32x � �32�

� 5.7

ExercisesExercises

Find the geometric mean between each pair of numbers.

1. 4 and 4 2. 4 and 6

3. 6 and 9 4. �12� and 2

5. 2�3� and 3�3� 6. 4 and 25

7. �3� and �6� 8. 10 and 100

9. �12� and �

14� 10. and

11. 4 and 16 12. 3 and 24

The geometric mean and one extreme are given. Find the other extreme.

13. �24� is the geometric mean between a and b. Find b if a � 2.

14. �12� is the geometric mean between a and b. Find b if a � 3.

Determine whether each statement is always, sometimes, or never true.

15. The geometric mean of two positive numbers is greater than the average of the twonumbers.

16. If the geometric mean of two positive numbers is less than 1, then both of the numbersare less than 1.

3�2��5

2�2��5

ExampleExample


Altitude of a Triangle In the diagram, �ABC � �ADB � �BDC.An altitude to the hypotenuse of a right triangle forms two right triangles. The two triangles are similar and each is similar to the original triangle. CD

B

A

Study Guide and Intervention (continued)

Geometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

Use right �ABC with B�D� ⊥ A�C�. Describe two geometricmeans.

a. �ADB � �BDC so �BAD

D� � �BC

DD�.

In �ABC, the altitude is the geometricmean between the two segments of thehypotenuse.

b. �ABC � �ADB and �ABC � �BDC,

so �AACB� � �A

ADB� and �B

ACC� � �D

BCC�.

In �ABC, each leg is the geometricmean between the hypotenuse and thesegment of the hypotenuse adjacent tothat leg.

Find x, y, and z.

�PP

QR� � �

PP

QS�

�21

55� � �

1x5� PR � 25, PQ � 15, PS � x

25x � 225 Cross multiply.

x � 9 Divide each side by 25.

Theny � PR � SP

� 25 � 9� 16

�QPR

R� � �QR

RS�

�2z5� � �y

z� PR � 25, QR � z, RS � y

�2z5� � �1

z6� y � 16

z2 � 400 Cross multiply.

z � 20 Take the square root of each side.

z

y

x

15

R

Q P

S25

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find x, y, and z to the nearest tenth.

1. 2. 3.

4. 5. 6.x zy

62

x

z y

2

2xy

1

��3

��12

zxy

81

z

xy 5

2

x

1 3

Skills PracticeGeometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1


Less

on

7-1

Find the geometric mean between each pair of numbers. State exact answers andanswers to the nearest tenth.

1. 2 and 8 2. 9 and 36 3. 4 and 7

4. 5 and 10 5. 2�2� and 5�2� 6. 3�5� and 5�5�

Find the measure of each altitude. State exact answers and answers to the nearesttenth.

7. 8.

9. 10.

Find x and y.

11. 12.

13. 14.

2

5y

x

15

4

y

x

10

4

yx

3 9

yx

R T

S

U4.5 8G

E H

F

2

9

L

M

N

P 2

12

C

D

B

A 2

7


Find the geometric mean between each pair of numbers to the nearest tenth.

1. 8 and 12 2. 3�7� and 6�7� 3. �45� and 2

Find the measure of each altitude. State exact answers and answers to the nearesttenth.

4. 5.

Find x, y, and z.

6. 7.

8. 9.

10. CIVIL ENGINEERING An airport, a factory, and a shopping center are at the vertices of aright triangle formed by three highways. The airport and factory are 6.0 miles apart. Theirdistances from the shopping center are 3.6 miles and 4.8 miles, respectively. A service roadwill be constructed from the shopping center to the highway that connects the airport andfactory. What is the shortest possible length for the service road? Round to the nearesthundredth.

x y

10z

20x

y

2

3

z

zx y

625

23

z

xy

8

17

6

KL

J M

125

U

T A V

Practice Geometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

Reading to Learn MathematicsGeometric Mean

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1


Less

on

7-1

Pre-Activity How can the geometric mean be used to view paintings?

Read the introduction to Lesson 7-1 at the top of page 342 in your textbook.

• What is a disadvantage of standing too close to a painting?

• What is a disadvantage of standing too far from a painting?

Reading the Lesson1. In the past, when you have seen the word mean in mathematics, it referred to the

average or arithmetic mean of the two numbers.

a. Complete the following by writing an algebraic expression in each blank.

If a and b are two positive numbers, then the geometric mean between a and b is

and their arithmetic mean is .

b. Explain in words, without using any mathematical symbols, the difference betweenthe geometric mean and the algebraic mean.

2. Let r and s be two positive numbers. In which of the following equations is z equal to thegeometric mean between r and s?

A. �zs

� � �zr� B. �z

r� � �z

s� C. s : z � z: r D. �z

r� � �

zs� E. �

zr� � �

zs� F. �

zs� � �z

r�

3. Supply the missing words or phrases to complete the statement of each theorem.

a. The measure of the altitude drawn from the vertex of the right angle of a right triangle

to its hypotenuse is the between the measures of the two

segments of the .

b. If the altitude is drawn from the vertex of the angle of a right

triangle to its hypotenuse, then the measure of a of the triangle

is the between the measure of the hypotenuse and the segment

of the adjacent to that leg.

c. If the altitude is drawn from the of the right angle of a right

triangle to its , then the two triangles formed are

to the given triangle and to each other.

Helping You Remember4. A good way to remember a new mathematical concept is to relate it to something you

already know. How can the meaning of mean in a proportion help you to remember howto find the geometric mean between two numbers?


Mathematics and MusicPythagoras, a Greek philosopher who lived during the sixth century B.C.,believed that all nature, beauty, and harmony could be expressed by whole-number relationships. Most people remember Pythagoras for his teachingsabout right triangles. (The sum of the squares of the legs equals the square ofthe hypotenuse.) But Pythagoras also discovered relationships between themusical notes of a scale. These relationships can be expressed as ratios.

C D E F G A B C�

�11� �

89� �

45� �

34� �

23� �

35� �1

85� �

12�

When you play a stringed instrument, The C string can be usedyou produce different notes by placing to produce F by placingyour finger on different places on a string. a finger �

34� of the way

This is the result of changing the lengthalong the string.of the vibrating part of the string.

Suppose a C string has a length of 16 inches. Write and solve proportions to determine what length of string would have to vibrate to produce the remaining notes of the scale.

1. D 2. E 3. F

4. G 5. A 6. B

7. C�

8. Complete to show the distance between finger positions on the 16-inch

C string for each note. For example, C(16) � D�14�29�� 1�

79�.

C D E F G A B C�

9. Between two consecutive musical notes, there is either a whole step or a half step. Using the distances you found in Exercise 8, determine what two pairs of notes have a half step between them.

1�79� in.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-17-1

34 of C string

Study Guide and InterventionThe Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

The Pythagorean Theorem In a right triangle, the sum of the squares of the measures of the legs equals the square of the measure of the hypotenuse.

�ABC is a right triangle, so a2 � b2 � c2.

Prove the Pythagorean Theorem.With altitude C�D�, each leg a and b is a geometric mean between hypotenuse c and the segment of the hypotenuse adjacent to that leg.

�ac

� � �ay� and �b

c� � �

bx�, so a2 � cy and b2 � cx.

Add the two equations and substitute c � y � x to geta2 � b2 � cy � cx � c( y � x) � c2.

c y

x a

b

hA C

BD

c a

bA C

B

Example 1Example 1

Example 2Example 2

a. Find a.

a2 � b2 � c2 Pythagorean Theorem

a2 � 122 � 132 b � 12, c � 13

a2 � 144 � 169 Simplify.a2 � 25 Subtract.

a � 5 Take the square root of each side.

a

12

13

AC

B

b. Find c.

a2 � b2 � c2 Pythagorean Theorem

202 � 302 � c2 a � 20, b � 30

400 � 900 � c2 Simplify.

1300 � c2 Add.

�1300� � c Take the square root of each side.

36.1 � c Use a calculator.

c

30

20

AC

B

ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6.x

1128

x

33

16x

59

49

x

6525

x

159

x

3 3


Converse of the Pythagorean Theorem If the sum of the squares of the measures of the two shorter sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

If the three whole numbers a, b, and c satisfy the equation a2 � b2 � c2, then the numbers a, b, and c form a If a2 � b2 � c2, then

Pythagorean triple. �ABC is a right triangle.

Determine whether �PQR is a right triangle.a2 � b2 � c2 Pythagorean Theorem

102 � (10�3�)2 � 202 a � 10, b � 10�3�, c � 20

100 � 300 � 400 Simplify.

400 � 400✓ Add.

The sum of the squares of the two shorter sides equals the square of the longest side, so thetriangle is a right triangle.

Determine whether each set of measures can be the measures of the sides of aright triangle. Then state whether they form a Pythagorean triple.

1. 30, 40, 50 2. 20, 30, 40 3. 18, 24, 30

4. 6, 8, 9 5. �37�, �

47�, �

57� 6. 10, 15, 20

7. �5�, �12�, �13� 8. 2, �8�, �12� 9. 9, 40, 41

A family of Pythagorean triples consists of multiples of known triples. For eachPythagorean triple, find two triples in the same family.

10. 3, 4, 5 11. 5, 12, 13 12. 7, 24, 25

10��3

20 10

QR

P

c

ab

A

C

B


The Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

ExampleExample

ExercisesExercises

Skills PracticeThe Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

Find x.

1. 2. 3.

4. 5. 6.

Determine whether �STU is a right triangle for the given vertices. Explain.

7. S(5, 5), T(7, 3), U(3, 2) 8. S(3, 3), T(5, 5), U(6, 0)

9. S(4, 6), T(9, 1), U(1, 3) 10. S(0, 3), T(�2, 5), U(4, 7)

11. S(�3, 2), T(2, 7), U(�1, 1) 12. S(2, �1), T(5, 4), U(6, �3)


13. 12, 16, 20 14. 16, 30, 32 15. 14, 48, 50

16. �25�, �

45�, �

65� 17. 2�6�, 5, 7 18. 2�2�, 2�7�, 6

x

31

14x9 9

8

x12.5

25

x

1232

x

12

13x

12

9


Find x.

1. 2. 3.

4. 5. 6.

Determine whether �GHI is a right triangle for the given vertices. Explain.

7. G(2, 7), H(3, 6), I(�4, �1) 8. G(�6, 2), H(1, 12), I(�2, 1)

9. G(�2, 1), H(3, �1), I(�4, �4) 10. G(�2, 4), H(4, 1), I(�1, �9)


11. 9, 40, 41 12. 7, 28, 29 13. 24, 32, 40

14. �95�, �

152�, 3 15. �

13�, , 1 16. , , �

47�

17. CONSTRUCTION The bottom end of a ramp at a warehouse is 10 feet from the base of the main dock and is 11 feet long. How high is the dock?

11 ft?

dock

ramp

10 ft

2�3��7

�4��7

2�2��3

x2424

42

x16

14

x

34

22

x26

2618

x

34 21x

13

23

Practice The Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

Reading to Learn MathematicsThe Pythagorean Theorem and Its Converse

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2


Less

on

7-2

Pre-Activity How are right triangles used to build suspension bridges?


Do the two right triangles shown in the drawing appear to be similar?Explain your reasoning.

Reading the Lesson

1. Explain in your own words the difference between how the Pythagorean Theorem is usedand how the Converse of the Pythagorean Theorem is used.

2. Refer to the figure. For this figure, which statements are true?

A. m2 � n2 � p2 B. n2 � m2 � p2

C. m2 � n2 � p2 D. m2 � p2 � n2

E. p2 � n2 � m2 F. n2 � p2 � m2

G. n � �m2 ��p2� H. p � �m2 ��n2�

3. Is the following statement true or false?A Pythagorean triple is any group of three numbers for which the sum of the squares of thesmaller two numbers is equal to the square of the largest number. Explain your reasoning.

4. If x, y, and z form a Pythagorean triple and k is a positive integer, which of the followinggroups of numbers are also Pythagorean triples?

A. 3x, 4y, 5z B. 3x, 3y, 3z C. �3x, �3y, �3z D. kx, ky, kz

Helping You Remember

5. Many students who studied geometry long ago remember the Pythagorean Theorem as theequation a2 � b2 � c2, but cannot tell you what this equation means. A formula is uselessif you don’t know what it means and how to use it. How could you help someone who hasforgotten the Pythagorean Theorem remember the meaning of the equation a2 � b2 � c2?

pm

n


Converse of a Right Triangle TheoremYou have learned that the measure of the altitude from the vertex ofthe right angle of a right triangle to its hypotenuse is the geometricmean between the measures of the two segments of the hypotenuse.Is the converse of this theorem true? In order to find out, it will helpto rewrite the original theorem in if-then form as follows.

If �ABQ is a right triangle with right angle at Q, then QP is the geometric mean between AP and PB, where Pis between A and B and Q�P� is perpendicular to A�B�.

1. Write the converse of the if-then form of the theorem.

2. Is the converse of the original theorem true? Refer to the figure at the right to explain your answer.

You may find it interesting to examine the other theorems inChapter 7 to see whether their converses are true or false. You willneed to restate the theorems carefully in order to write theirconverses.

Q

BP

A

Q

BPA

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-27-2

Study Guide and InterventionSpecial Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3

Properties of 45°-45°-90° Triangles The sides of a 45°-45°-90° right triangle have aspecial relationship.

If the leg of a 45°-45°-90°right triangle is x units, show that the hypotenuse is x�2� units.

Using the Pythagorean Theorem with a � b � x, then

c2 � a2 � b2

� x2 � x2

� 2x2

c � �2x2�� x�2�

x��

x

x 245�

45�

In a 45°-45°-90° right triangle the hypotenuse is �2� times the leg. If the hypotenuse is 6 units,find the length of each leg.The hypotenuse is �2� times the leg, sodivide the length of the hypotenuse by �2�.

a �

�

�

� 3�2� units

6�2��2

6�2��2��2�

6��2�


ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6.

7. Find the perimeter of a square with diagonal 12 centimeters.

8. Find the diagonal of a square with perimeter 20 inches.

9. Find the diagonal of a square with perimeter 28 meters.

x 3��2x 18x x

18

x10x

45�

3��2x

8

45�

45�


Properties of 30°-60°-90° Triangles The sides of a 30°-60°-90° right triangle alsohave a special relationship.

In a 30°-60°-90° right triangle, show that the hypotenuse is twice the shorter leg and the longer leg is �3� times the shorter leg.

�MNQ is a 30°-60°-90° right triangle, and the length of the hypotenuse M�N� is two times the length of the shorter side N�Q�.Using the Pythagorean Theorem,a2 � (2x) 2 � x2

� 4x2 � x2

� 3x2

a � �3x2�� x�3�

In a 30°-60°-90° right triangle, the hypotenuse is 5 centimeters.Find the lengths of the other two sides of the triangle.If the hypotenuse of a 30°-60°-90° right triangle is 5 centimeters, then the length of theshorter leg is half of 5 or 2.5 centimeters. The length of the longer leg is �3� times the length of the shorter leg, or (2.5)(�3�) centimeters.

Find x and y.

1. 2. 3.

4. 5. 6.

7. The perimeter of an equilateral triangle is 32 centimeters. Find the length of an altitudeof the triangle to the nearest tenth of a centimeter.

8. An altitude of an equilateral triangle is 8.3 meters. Find the perimeter of the triangle tothe nearest tenth of a meter.

xy

60�

20

xy60�

12

xy

30�

9��3

x

y

11

30�

x

y

60�

8

x

y30�

60�12

x

a

N

Q

P

M

2x

30�30�

60�

60�


Special Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

ExercisesExercises

Example 1Example 1

Example 2Example 2

�MNP is an equilateraltriangle.

�MNQ is a 30°-60°-90°right triangle.

Skills PracticeSpecial Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3

Find x and y.

1. 2. 3.

4. 5. 6.

For Exercises 7–9, use the figure at the right.

7. If a � 11, find b and c.

8. If b � 15, find a and c.

9. If c � 9, find a and b.

For Exercises 10 and 11, use the figure at the right.

10. The perimeter of the square is 30 inches. Find the length of B�C�.

11. Find the length of the diagonal B�D�.

12. The perimeter of the equilateral triangle is 60 meters. Find the length of an altitude.

13. �GEC is a 30°-60°-90° triangle with right angle at E, and E�C� is the longer leg. Find the coordinates of G in Quadrant I for E(1, 1) and C(4, 1).

E

FGD 60�

A B

CD 45�

bA

B

C

ac

60�

30�

y

x�

13

1313

13

y

x60�

16

y

x

45� 8

y

x

45�

12

y

x

30�

32

y

x60� 24


Find x and y.

1. 2. 3.

4. 5. 6.

For Exercises 7–9, use the figure at the right.

7. If a � 4�3�, find b and c.

8. If x � 3�3�, find a and CD.

9. If a � 4, find CD, b, and y.

10. The perimeter of an equilateral triangle is 39 centimeters. Find the length of an altitudeof the triangle.

11. �MIP is a 30°-60°-90° triangle with right angle at I, and I�P� the longer leg. Find thecoordinates of M in Quadrant I for I(3, 3) and P(12, 3).

12. �TJK is a 45°-45°-90° triangle with right angle at J. Find the coordinates of T inQuadrant II for J(�2, �3) and K(3, �3).

13. BOTANICAL GARDENS One of the displays at a botanical garden is an herb garden planted in the shape of a square. The square measures 6 yards on each side. Visitors can view the herbs from adiagonal pathway through the garden. How long is the pathway?

6 yd 6 yd

6 yd

6 yd

bA

B

C

D

a

x

y60�

30�

c

x

45�

11

y60�3.5

xy

x�

y 28

y

x

30�

26y

x

25

60�

yx

45�9

Practice Special Right Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

Reading to Learn MathematicsSpecial Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3


Less

on

7-3

Pre-Activity How is triangle tiling used in wallpaper design?

Read the introduction to Lesson 7-3 at the top of page 357 in your textbook.• How can you most completely describe the larger triangle and the two

smaller triangles in tile 15?

• How can you most completely describe the larger triangle and the twosmaller triangles in tile 16? (Include angle measures in describing all thetriangles.)

Reading the Lesson1. Supply the correct number or numbers to complete each statement.

a. In a 45°-45°-90° triangle, to find the length of the hypotenuse, multiply the length of a

leg by .

b. In a 30°-60°-90° triangle, to find the length of the hypotenuse, multiply the length of

the shorter leg by .

c. In a 30°-60°-90° triangle, the longer leg is opposite the angle with a measure of .

d. In a 30°-60°-90° triangle, to find the length of the longer leg, multiply the length of

the shorter leg by .

e. In an isosceles right triangle, each leg is opposite an angle with a measure of .

f. In a 30°-60°-90° triangle, to find the length of the shorter leg, divide the length of the

longer leg by .

g. In 30°-60°-90° triangle, to find the length of the longer leg, divide the length of the

hypotenuse by and multiply the result by .

h. To find the length of a side of a square, divide the length of the diagonal by .

2. Indicate whether each statement is always, sometimes, or never true.a. The lengths of the three sides of an isosceles triangle satisfy the Pythagorean

Theorem.b. The lengths of the sides of a 30°-60°-90° triangle form a Pythagorean triple.c. The lengths of all three sides of a 30°-60°-90° triangle are positive integers.

Helping You Remember3. Some students find it easier to remember mathematical concepts in terms of specific

numbers rather than variables. How can you use specific numbers to help you rememberthe relationship between the lengths of the three sides in a 30°-60°-90° triangle?


Constructing Values of Square RootsThe diagram at the right shows a right isosceles triangle with two legs of length 1 inch. By the Pythagorean Theorem, the length of the hypotenuse is �2� inches. By constructing an adjacent right triangle with legs of �2� inches and 1 inch, you can create a segment of length �3�.

By continuing this process as shown below, you can construct a “wheel” of square roots. This wheel is called the “Wheel of Theodorus”after a Greek philosopher who lived about 400 B.C.

Continue constructing the wheel until you make a segment oflength �18�.

��

1

1

1

3��

2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-37-3

1

1

11

1

1

��2

��3

��5

��6

��7

��8

��10

��11 ��12��13

��14

��15

��17

��18

��16 � 4

��4 � 2

��9 � 3

Study Guide and InterventionTrigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Trigonometric Ratios The ratio of the lengths of two sides of a right triangle is called a trigonometric ratio. The three most common ratios are sine, cosine, and tangent, which are abbreviated sin, cos, and tan,respectively.

sin R � �leg

hyopppotoesnitues�

eR

� cos R � tan R �

� �rt� � �

st� � �

rs�

Find sin A, cos A, and tan A. Express each ratio as a decimal to the nearest thousandth.

sin A � �ohpyppoostietneulseeg

� cos A � �ahdyjpaocteenntulseeg

� tan A � �aopd

pja

ocseintet

lleegg�

� �BAB

C� � �A

ABC� � �

BAC

C�

� �153� � �

11

23� � �1

52�

� 0.385 � 0.923 � 0.417

Find the indicated trigonometric ratio as a fraction and as a decimal. If necessary, round to the nearest ten-thousandth.

1. sin A 2. tan B

3. cos A 4. cos B

5. sin D 6. tan E

7. cos E 8. cos D

16

1620

12

3430

C

B

A D F

E

12

135

C

B

A

leg opposite �R��leg adjacent to �R

leg adjacent to �R��hypotenuse

s

tr

T

S

R

ExercisesExercises

ExampleExample


Use Trigonometric Ratios In a right triangle, if you know the measures of two sidesor if you know the measures of one side and an acute angle, then you can use trigonometricratios to find the measures of the missing sides or angles of the triangle.

Find x, y, and z. Round each measure to the nearest whole number. 1858�

x � CB y

zA


Trigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

a. Find x.

x � 58 � 90x � 32

b. Find y.

tan A � �1y8�

tan 58° � �1y8�

y � 18 tan 58°y � 29

c. Find z.

cos A � �1z8�

cos 58° � �1z8�

z cos 58° � 18

z � �cos18

58°�

z � 34

ExercisesExercises

Find x. Round to the nearest tenth.

1. 2.

3. 4.

5. 6.15

64� x16

40�

x

4

1x�12

5x�

12 16

x�3228�

x

ExampleExample

Skills PracticeTrigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Use �RST to find sin R, cos R, tan R, sin S, cos S, and tan S.Express each ratio as a fraction and as a decimal to the nearest hundredth.

1. r � 16, s � 30, t � 34 2. r � 10, s � 24, t � 26

Use a calculator to find each value. Round to the nearest ten-thousandth.

3. sin 5 4. tan 23 5. cos 61

6. sin 75.8 7. tan 17.3 8. cos 52.9

Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to thenearest ten-thousandth.

9. tan C 10. sin A 11. cos C

Find the measure of each acute angle to the nearest tenth of a degree.

12. sin B � 0.2985 13. tan A � 0.4168 14. cos R � 0.8443

15. tan C � 0.3894 16. cos B � 0.7329 17. sin A � 0.1176


18. 19. 20.

19

x

33� UL

S

27

x �

8

BA

C

27

x �

13

BA

C

41

409B

A

C

sR

S

T

rt


Use �LMN to find sin L, cos L, tan L, sin M, cos M, and tan M.Express each ratio as a fraction and as a decimal to the nearest hundredth.

1. � � 15, m � 36, n� 39 2. � � 12, m � 12�3�, n � 24

Use a calculator to find each value. Round to the nearest ten-thousandth.

3. sin 92.4 4. tan 27.5 5. cos 64.8

Use the figure to find each trigonometric ratio. Express answers as a fraction and as a decimal rounded to the nearest ten-thousandth.

6. cos A 7. tan B 8. sin A

Find the measure of each acute angle to the nearest tenth of a degree.

9. sin B � 0.7823 10. tan A � 0.2356 11. cos R � 0.6401


12. 13. 14.

15. GEOGRAPHY Diego used a theodolite to map a region of land for his class in geomorphology. To determine the elevation of a vertical rockformation, he measured the distance from the base of the formation to his position and the angle between the ground and the line of sight to the top of the formation. The distance was 43 meters and the angle was 36 degrees. What is the height of the formation to the nearest meter?

36�

43 m

41�x

3229

x �9

23

x �

11

15

5��105

CA

B

ML

N

Practice Trigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

Reading to Learn MathematicsTrigonometry

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4


Less

on

7-4

Pre-Activity How can surveyors determine angle measures?


• Why is it important to determine the relative positions accurately innavigation? (Give two possible reasons.)

• What does calibrated mean?

Reading the Lesson

1. Refer to the figure. Write a ratio using the side lengths in the figure to represent each of the following trigonometric ratios.

A. sin N B. cos N

C. tan N D. tan M

E. sin M F. cos M

2. Assume that you enter each of the expressions in the list on the left into your calculator.Match each of these expressions with a description from the list on the right to tell whatyou are finding when you enter this expression.

P

M N

a. sin 20

b. cos 20

c. sin�1 0.8

d. tan�1 0.8

e. tan 20

f. cos�1 0.8

i. the degree measure of an acute angle whose cosine is 0.8

ii. the ratio of the length of the leg adjacent to the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle

iii.the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe adjacent leg is 0.8

iv. the ratio of the length of the leg opposite the 20° angle to thelength of the leg adjacent to it in a 20°-70°-90° triangle

v. the ratio of the length of the leg opposite the 20° angle to thelength of hypotenuse in a 20°-70°-90° triangle

vi. the degree measure of an acute angle in a right triangle for which the ratio of the length of the opposite leg to the length ofthe hypotenuse is 0.8


3. How can the co in cosine help you to remember the relationship between the sines andcosines of the two acute angles of a right triangle?


Sine and Cosine of AnglesThe following diagram can be used to obtain approximate values for the sineand cosine of angles from 0° to 90°. The radius of the circle is 1. So, the sineand cosine values can be read directly from the vertical and horizontal axes.

Find approximate values for sin 40°and cos 40�. Consider the triangle formed by the segment marked 40°, as illustrated by the shaded triangle at right.

sin 40° � �ac� � �

0.164� or 0.64 cos 40° � �

bc� � �

0.177� or 0.77

1. Use the diagram above to complete the chart of values.

2. Compare the sine and cosine of two complementary angles (angles whose sum is 90°). What do you notice?

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

90°

0°

10°

20°

30°

40°

50°

60°

70°80°

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-47-4

x° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90°

sin x° 0.64

cos x° 0.77

1

0

40°0.64

c � 1 unit

x °b � cos x ° 0.77 1

a � sin x °

ExampleExample

Study Guide and InterventionAngles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Angles of Elevation Many real-world problems that involve looking up to an object can be described in terms of an angle of elevation, which is the angle between an observer’s line of sight and a horizontal line.

The angle of elevation from point A to the top of a cliff is 34°. If point A is 1000 feet from the base of the cliff,how high is the cliff?Let x � the height of the cliff.

tan 34° � �10x00� tan � �

oapdpjaocseitnet

�

1000(tan 34°) � x Multiply each side by 1000.

674.5 � x Use a calculator.

The height of the cliff is about 674.5 feet.

Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.

1. The angle of elevation from point A to the top of a hill is 49°.If point A is 400 feet from the base of the hill, how high is the hill?

2. Find the angle of elevation of the sun when a 12.5-meter-tall telephone pole casts a 18-meter-long shadow.

3. A ladder leaning against a building makes an angle of 78°with the ground. The foot of the ladder is 5 feet from the building. How long is the ladder?

4. A person whose eyes are 5 feet above the ground is standing on the runway of an airport 100 feet from the control tower.That person observes an air traffic controller at the window of the 132-foot tower. What is the angle of elevation?

?5 ft

100 ft

132 ft

78�5 ft

?

18 m

12.5 m

sun

?

✹

400 ft

?

49�A

?

1000 ft34�A

x

angle ofelevation

line of si

ght

ExercisesExercises

ExampleExample


Angles of Depression When an observer is looking down, the angle of depression is the angle between the observer’s line of sight and a horizontal line.

The angle of depression from the top of an 80-foot building to point A on the ground is 42°. How far is the foot of the building from point A?Let x � the distance from point A to the foot of the building. Since the horizontal line is parallel to the ground, the angle of depression�DBA is congruent to �BAC.

tan 42° � �8x0� tan � �

o

a

p

d

p

ja

o

c

s

e

it

n

e

t�

x(tan 42°) � 80 Multiply each side by x.

x � �tan80

42°� Divide each side by tan 42°.

x � 88.8 Use a calculator.

Point A is about 89 feet from the base of the building.

Solve each problem. Round measures of segments to the nearest whole numberand angles to the nearest degree.

1. The angle of depression from the top of a sheer cliff to point A on the ground is 35°. If point A is 280 feet from the base of the cliff, how tall is the cliff?

2. The angle of depression from a balloon on a 75-foot string to a person on the ground is 36°. How high is the balloon?

3. A ski run is 1000 yards long with a vertical drop of 208 yards. Find the angle of depression from the top of the ski run to the bottom.

4. From the top of a 120-foot-high tower, an air traffic controller observes an airplane on the runway at an angle of depression of 19°. How far from the base of thetower is the airplane?

120 ft

?

19�

208 yd

?

1000 yd

36�

75 ft ?

A

35�

280 ft

?

A C

BD

x42�

angle ofdepression

horizontal

80 ft

Yline of sight

horizontalangle ofdepression


Angles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

ExercisesExercises

ExampleExample

Skills PracticeAngles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Name the angle of depression or angle of elevation in each figure.

1. 2.

3. 4.

5. MOUNTAIN BIKING On a mountain bike trip along the Gemini Bridges Trail in Moab,Utah, Nabuko stopped on the canyon floor to get a good view of the twin sandstonebridges. Nabuko is standing about 60 meters from the base of the canyon cliff, and thenatural arch bridges are about 100 meters up the canyon wall. If her line of sight is fivefeet above the ground, what is the angle of elevation to the top of the bridges? Round tothe nearest tenth degree.

6. SHADOWS Suppose the sun casts a shadow off a 35-foot building.If the angle of elevation to the sun is 60°, how long is the shadow to the nearest tenth of a foot?

7. BALLOONING From her position in a hot-air balloon, Angie can see her car parked in afield. If the angle of depression is 8° and Angie is 38 meters above the ground, what isthe straight-line distance from Angie to her car? Round to the nearest whole meter.

8. INDIRECT MEASUREMENT Kyle is at the end of a pier 30 feet above the ocean. His eye level is 3 feet above the pier. He is using binoculars to watch a whale surface. If the angle of depression of the whale is 20°, how far is the whale from Kyle’s binoculars? Round to the nearest tenth foot.

whale water level

20�Kyle’s eyes

pier3 ft

30 ft

60�?

35 ft

Z

P

W

R

D

A

C

B

T

W

R

S

F

T

L

S


Name the angle of depression or angle of elevation in each figure.

1. 2.

3. WATER TOWERS A student can see a water tower from the closest point of the soccerfield at San Lobos High School. The edge of the soccer field is about 110 feet from thewater tower and the water tower stands at a height of 32.5 feet. What is the angle ofelevation if the eye level of the student viewing the tower from the edge of the soccerfield is 6 feet above the ground? Round to the nearest tenth degree.

4. CONSTRUCTION A roofer props a ladder against a wall so that the top of the ladderreaches a 30-foot roof that needs repair. If the angle of elevation from the bottom of theladder to the roof is 55°, how far is the ladder from the base of the wall? Round youranswer to the nearest foot.

5. TOWN ORDINANCES The town of Belmont restricts the height of flagpoles to 25 feet on any property. Lindsay wants to determinewhether her school is in compliance with the regulation. Her eye level is 5.5 feet from the ground and she stands 36 feet from theflagpole. If the angle of elevation is about 25°, what is the height of the flagpole to the nearest tenth foot?

6. GEOGRAPHY Stephan is standing on a mesa at the Painted Desert. The elevation ofthe mesa is about 1380 meters and Stephan’s eye level is 1.8 meters above ground. IfStephan can see a band of multicolored shale at the bottom and the angle of depressionis 29°, about how far is the band of shale from his eyes? Round to the nearest meter.

7. INDIRECT MEASUREMENT Mr. Dominguez is standing on a 40-foot ocean bluff near his home. He can see his two dogs on the beach below. If his line of sight is 6 feet above the ground and the angles of depression to his dogs are 34°and 48°, how far apart are the dogs to the nearest foot?

48� 34�

40 ft

6 ft

Mr. Dominguez

bluff

25�5.5 ft

36 ft

x

R

M

P

L

T

Y

R

Z

Practice Angles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

Reading to Learn MathematicsAngles of Elevation and Depression

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5


Less

on

7-5

Pre-Activity How do airline pilots use angles of elevation and depression?


What does the angle measure tell the pilot?

Reading the Lesson

1. Refer to the figure. The two observers are looking at one another. Select the correct choice for each question.

a. What is the line of sight?(i) line RS (ii) line ST (iii) line RT (iv) line TU

b. What is the angle of elevation?(i) �RST (ii) �SRT (iii) �RTS (iv) �UTR

c. What is the angle of depression?(i) �RST (ii) �SRT (iii) �RTS (iv) �UTR

d. How are the angle of elevation and the angle of depression related?(i) They are complementary.(ii) They are congruent.(iii) They are supplementary.(iv) The angle of elevation is larger than the angle of depression.

e. Which postulate or theorem that you learned in Chapter 3 supports your answer forpart c?(i) Corresponding Angles Postulate(ii) Alternate Exterior Angles Theorem(iii) Consecutive Interior Angles Theorem(iv) Alternate Interior Angles Theorem

2. A student says that the angle of elevation from his eye to the top of a flagpole is 135°.What is wrong with the student’s statement?


3. A good way to remember something is to explain it to someone else. Suppose a classmatefinds it difficult to distinguish between angles of elevation and angles of depression. Whatare some hints you can give her to help her get it right every time?

S

Tobserver at

top of building

observeron ground R

U


Reading MathematicsThe three most common trigonometric ratios are sine, cosine, and tangent. Three other ratios are thecosecant, secant, and cotangent. The chart below shows abbreviations and definitions for all six ratios.Refer to the triangle at the right.

Use the abbreviations to rewrite each statement as an equation.

1. The secant of angle A is equal to 1 divided by the cosine of angle A.

2. The cosecant of angle A is equal to 1 divided by the sine of angle A.

3. The cotangent of angle A is equal to 1 divided by the tangent of angle A.

4. The cosecant of angle A multiplied by the sine of angle A is equal to 1.

5. The secant of angle A multiplied by the cosine of angle A is equal to 1.

6. The cotangent of angle A times the tangent of angle A is equal to 1.

Use the triangle at right. Write each ratio.

7. sec R 8. csc R 9. cot R

10. sec S 11. csc S 12. cot S

13. If sin x° � 0.289, find the value of csc x°.

14. If tan x° � 1.376, find the value of cot x°.

R

T S

ts

r

A

ca

b

B

C

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-57-5

Abbreviation Read as: Ratio

sin A the sine of �A � �ac

�

cos A the cosine of �A � �bc

�

tan A the tangent of �A � �ab

�

csc A the cosecant of �A � �ac

�

sec A the secant of �A � �bc

�

cot A the cotangent of �A � �ba

�leg adjacent to �A��

leg opposite �A

hypotenuse��leg adjacent to �A

hypotenuse��leg opposite �A

leg opposite �A��leg adjacent to �A

leg adjacent to �A��

hypotenuse

leg opposite �A��

hypotenuse

Study Guide and InterventionThe Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6


Less

on

7-6

The Law of Sines In any triangle, there is a special relationship between the angles ofthe triangle and the lengths of the sides opposite the angles.

Law of Sines �sin

aA

� � �sin

bB

� � �sin

cC

�

In �ABC, find b.

�sin

cC

� � �sin

bB

� Law of Sines

�sin

3045°� � �

sinb74°� m�C � 45, c � 30, m�B � 74

b sin 45° � 30 sin 74° Cross multiply.

b � �30

sisnin45

7°4°

� Divide each side by sin 45°.

b � 40.8 Use a calculator.

45�

3074�

b

B

AC

In �DEF, find m�D.

�sin

dD

� � �sin

eE

� Law of Sines

�si

2n8D

� � �sin

2458°�

d � 28, m�E � 58,

e � 24

24 sin D � 28 sin 58° Cross multiply.

sin D � �28 s

2in4

58°� Divide each side by 24.

D � sin�1 �28 s

2in4

58°� Use the inverse sine.

D � 81.6° Use a calculator.

58�

24

28

E

FD


ExercisesExercises

Find each measure using the given measures of �ABC. Round angle measures tothe nearest degree and side measures to the nearest tenth.

1. If c � 12, m�A � 80, and m�C � 40, find a.

2. If b � 20, c � 26, and m�C � 52, find m�B.

3. If a � 18, c � 16, and m�A � 84, find m�C.

4. If a � 25, m�A � 72, and m�B � 17, find b.

5. If b � 12, m�A � 89, and m�B� 80, find a.

6. If a � 30, c � 20, and m�A � 60, find m�C.


Use the Law of Sines to Solve Problems You can use the Law of Sines to solvesome problems that involve triangles.

Law of SinesLet �ABC be any triangle with a, b, and c representing the measures of the sides opposite

the angles with measures A, B, and C, respectively. Then �sina

A� � �sinb

B� � �sinc

C�.

Isosceles �ABC has a base of 24 centimeters and a vertex angle of 68°. Find the perimeter of the triangle.The vertex angle is 68°, so the sum of the measures of the base angles is 112 and m�A � m�C � 56.

�sin

bB

� � �sin

aA

� Law of Sines

�sin

2468°� � �

sina56°� m�B � 68, b � 24, m�A � 56

a sin 68° � 24 sin 56° Cross multiply.

a � �24

sisnin68

5°6°

� Divide each side by sin 68°.

� 21.5 Use a calculator.

The triangle is isosceles, so c � 21.5.The perimeter is 24 � 21.5 � 21.5 or about 67 centimeters.

Draw a triangle to go with each exercise and mark it with the given information.Then solve the problem. Round angle measures to the nearest degree and sidemeasures to the nearest tenth.

1. One side of a triangular garden is 42.0 feet. The angles on each end of this side measure66° and 82°. Find the length of fence needed to enclose the garden.

2. Two radar stations A and B are 32 miles apart. They locate an airplane X at the sametime. The three points form �XAB, which measures 46°, and �XBA, which measures52°. How far is the airplane from each station?

3. A civil engineer wants to determine the distances from points A and B to an inaccessiblepoint C in a river. �BAC measures 67° and �ABC measures 52°. If points A and B are82.0 feet apart, find the distance from C to each point.

4. A ranger tower at point A is 42 kilometers north of a ranger tower at point B. A fire atpoint C is observed from both towers. If �BAC measures 43° and �ABC measures 68°,which ranger tower is closer to the fire? How much closer?

68�

b

c a

24

B

CA


The Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6

ExampleExample

ExercisesExercises

Skills PracticeThe Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6


Less

on

7-6

Find each measure using the given measures from �ABC. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.

1. If m�A � 35, m�B � 48, and b � 28, find a.

2. If m�B � 17, m�C � 46, and c � 18, find b.

3. If m�C � 86, m�A � 51, and a � 38, find c.

4. If a � 17, b � 8, and m�A � 73, find m�B.

5. If c � 38, b � 34, and m�B � 36, find m�C.

6. If a � 12, c � 20, and m�C � 83, find m�A.

7. If m�A � 22, a � 18, and m�B� 104, find b.

Solve each �PQR described below. Round measures to the nearest tenth.

8. p � 27, q � 40, m�P � 33

9. q � 12, r � 11, m�R � 16

10. p � 29, q � 34, m�Q � 111

11. If m�P � 89, p � 16, r � 12

12. If m�Q � 103, m�P � 63, p � 13

13. If m�P � 96, m�R � 82, r � 35

14. If m�R � 49, m�Q � 76, r � 26

15. If m�Q � 31, m�P � 52, p � 20

16. If q � 8, m�Q � 28, m�R � 72

17. If r � 15, p � 21, m�P � 128


Find each measure using the given measures from �EFG. Round angle measuresto the nearest tenth degree and side measures to the nearest tenth.

1. If m�G � 14, m�E � 67, and e � 14, find g.

2. If e � 12.7, m�E � 42, and m�F � 61, find f.

3. If g � 14, f � 5.8, and m�G � 83, find m�F.

4. If e � 19.1, m�G � 34, and m�E � 56, find g.

5. If f � 9.6, g � 27.4, and m�G � 43, find m�F.

Solve each �STU described below. Round measures to the nearest tenth.

6. m�T � 85, s � 4.3, t � 8.2

7. s � 40, u � 12, m�S � 37

8. m�U � 37, t � 2.3, m�T � 17

9. m�S � 62, m�U � 59, s � 17.8

10. t � 28.4, u � 21.7, m�T � 66

11. m�S � 89, s � 15.3, t � 14

12. m�T � 98, m�U � 74, u � 9.6

13. t � 11.8, m�S � 84, m�T � 47

14. INDIRECT MEASUREMENT To find the distance from the edge of the lake to the tree on the island in the lake, Hannah set up atriangular configuration as shown in the diagram. The distance from location A to location B is 85 meters. The measures of the angles at A and B are 51° and 83°, respectively. What is the distancefrom the edge of the lake at B to the tree on the island at C?

A

C

B

Practice The Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6

Reading to Learn MathematicsThe Law of Sines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6


Less

on

7-6

Pre-Activity How are triangles used in radio astronomy?


Why might several antennas be better than one single antenna whenstudying distant objects?

Reading the Lesson

1. Refer to the figure. According to the Law of Sines, which of the following are correct statements?

A. �sinm

M� � �sinn

N� � �sinp

P� B. �sin

Mm

� � �si

Nn n� � �

sinP

p�

C. �co

ms M� � �

cosn

N� � �

cops P� D. �

sinm

M� � �

sinn

N� � �

sinp

P�

E. (sin M)2 � (sin N)2 � (sin P)2 F. �sin

pP

� � �sin

mM

� � �sin

nN

�

2. State whether each of the following statements is true or false. If the statement is false,explain why.

a. The Law of Sines applies to all triangles.

b. The Pythagorean Theorem applies to all triangles.

c. If you are given the length of one side of a triangle and the measures of any twoangles, you can use the Law of Sines to find the lengths of the other two sides.

d. If you know the measures of two angles of a triangle, you should use the Law of Sinesto find the measure of the third angle.

e. A friend tells you that in triangle RST, m�R � 132, r � 24 centimeters, and s � 31centimeters. Can you use the Law of Sines to solve the triangle? Explain.


3. Many students remember mathematical equations and formulas better if they can statethem in words. State the Law of Sines in your own words without using variables ormathematical symbols.

P

M Np

mn


IdentitiesAn identity is an equation that is true for all values of the variable for which both sides are defined. One way to verify an identity is to use a right triangle and the definitions fortrigonometric functions.

Verify that (sin A)2 � (cos A)2 � 1 is an identity.

(sin A)2 � (cos A)2 � ��ac��2 � ��

bc��2

� �a2 �

cb2

� � �cc

2

2� � 1

To check whether an equation may be an identity, you can testseveral values. However, since you cannot test all values, youcannot be certain that the equation is an identity.

Test sin 2x � 2 sin x cos x to see if it could be an identity.

Try x � 20. Use a calculator to evaluate each expression.

sin 2x � sin 40 2 sin x cos x � 2 (sin 20)(cos 20)� 0.643 � 2(0.342)(0.940)

� 0.643

Since the left and right sides seem equal, the equation may be an identity.

Use triangle ABC shown above. Verify that each equation is an identity.

1. �csoins

AA

� � �tan1

A� 2. �tsainn

BB

� � �co1s B�

3. tan B cos B � sin B 4. 1 � (cos B)2 � (sin B)2

Try several values for x to test whether each equation could be an identity.

5. cos 2x � (cos x)2 � (sin x)2 6. cos (90 � x) � sin x

B

A C

ca

b

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-67-6

Example 1Example 1

Example 2Example 2

Study Guide and InterventionThe Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7


Less

on

7-7

The Law of Cosines Another relationship between the sides and angles of any triangleis called the Law of Cosines. You can use the Law of Cosines if you know three sides of atriangle or if you know two sides and the included angle of a triangle.

Let �ABC be any triangle with a, b, and c representing the measures of the sides opposite Law of Cosines the angles with measures A, B, and C, respectively. Then the following equations are true.

a2 � b2 � c2 � 2bc cos A b2 � a2 � c2 � 2ac cos B c2 � a2 � b2 � 2ab cos C

In �ABC, find c.c2 � a2 � b2 � 2ab cos C Law of Cosines

c2 � 122 � 102 � 2(12)(10)cos 48° a � 12, b � 10, m�C � 48

c � �122 �� 102 �� 2(12)�(10)co�s 48°� Take the square root of each side.

c � 9.1 Use a calculator.

In �ABC, find m�A.a2 � b2 � c2 � 2bc cos A Law of Cosines

72 � 52 � 82 � 2(5)(8) cos A a � 7, b � 5, c � 8

49 � 25 � 64 � 80 cos A Multiply.

�40 � �80 cos A Subtract 89 from each side.

�12� � cos A Divide each side by �80.

cos�1 �12� � A Use the inverse cosine.

60° � A Use a calculator.

Find each measure using the given measures from �ABC. Round angle measuresto the nearest degree and side measures to the nearest tenth.

1. If b � 14, c � 12, and m�A � 62, find a.

2. If a � 11, b � 10, and c � 12, find m�B.

3. If a � 24, b � 18, and c � 16, find m�C.

4. If a � 20, c � 25, and m�B � 82, find b.

5. If b � 18, c � 28, and m�A � 59, find a.

6. If a � 15, b � 19, and c � 15, find m�C.

58

7 CB

A

48�12 10

c

C

BA

Example 1Example 1

Example 2Example 2

ExercisesExercises


Use the Law of Cosines to Solve Problems You can use the Law of Cosines tosolve some problems involving triangles.

Let �ABC be any triangle with a, b, and c representing the measures of the sides opposite the Law of Cosines angles with measures A, B, and C, respectively. Then the following equations are true.

a2 � b2 � c2 � 2bc cos A b2 � a2 � c2 � 2ac cos B c2 � a2 � b2 � 2ab cos C

Ms. Jones wants to purchase a piece of land with the shape shown. Find the perimeter of the property.Use the Law of Cosines to find the value of a.

a2 � b2 � c2 � 2bc cos A Law of Cosines

a2 � 3002 � 2002 � 2(300)(200) cos 88° b � 300, c � 200, m�A � 88

a � �130,0�00 ��120,0�00 cos� 88°� Take the square root of each side.


Use the Law of Cosines again to find the value of c.

c2 � a2 � b2 � 2ab cos C Law of Cosines

c2 � 354.72 � 3002 � 2(354.7)(300) cos 80° a � 354.7, b � 300, m�C � 80

c � �215,8�12.09� � 21�2,820� cos 8�0°� Take the square root of each side.


The perimeter of the land is 300 � 200 � 422.9 � 200 or about 1223 feet.

Draw a figure or diagram to go with each exercise and mark it with the giveninformation. Then solve the problem. Round angle measures to the nearest degreeand side measures to the nearest tenth.

1. A triangular garden has dimensions 54 feet, 48 feet, and 62 feet. Find the angles at eachcorner of the garden.

2. A parallelogram has a 68° angle and sides 8 and 12. Find the lengths of the diagonals.

3. An airplane is sighted from two locations, and its position forms an acute triangle withthem. The distance to the airplane is 20 miles from one location with an angle ofelevation 48.0°, and 40 miles from the other location with an angle of elevation of 21.8°.How far apart are the two locations?

4. A ranger tower at point A is directly north of a ranger tower at point B. A fire at point Cis observed from both towers. The distance from the fire to tower A is 60 miles, and thedistance from the fire to tower B is 50 miles. If m�ACB � 62, find the distance betweenthe towers.

200 ft

300 ft

300 ft

88�

80�ca


The Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7

ExampleExample

ExercisesExercises

Skills PracticeThe Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7


Less

on

7-7

In �RST, given the following measures, find the measure of the missing side.

1. r � 5, s � 8, m�T � 39

2. r � 6, t � 11, m�S � 87

3. r � 9, t � 15, m�S � 103

4. s � 12, t � 10, m�R � 58

In �HIJ, given the lengths of the sides, find the measure of the stated angle to thenearest tenth.

5. h � 12, i � 18, j � 7; m�H

6. h � 15, i � 16, j � 22; m�I

7. h � 23, i � 27, j � 29; m�J

8. h � 37, i � 21, j � 30; m�H

Determine whether the Law of Sines or the Law of Cosines should be used first tosolve each triangle. Then solve each triangle. Round angle measures to the nearestdegree and side measures to the nearest tenth.

9. 10.

11. a � 10, b � 14, c �19 12. a � 12, b � 10, m�C � 27

Solve each �RST described below. Round measures to the nearest tenth.

13. r � 12, s � 32, t � 34

14. r � 30, s � 25, m�T � 42

15. r � 15, s � 11, m�R � 67

16. r � 21, s � 28, t � 30

M

L N

�86�

52

24

B

A C

c

66�

33

19


In �JKL, given the following measures, find the measure of the missing side.

1. j � 1.3, k � 10, m�L � 77

2. j � 9.6, � � 1.7, m�K � 43

3. j � 11, k � 7, m�L � 63

4. k � 4.7, � � 5.2, m�J � 112

In �MNQ, given the lengths of the sides, find the measure of the stated angle tothe nearest tenth.

5. m � 17, n � 23, q � 25; m�Q

6. m � 24, n � 28, q � 34; m�M

7. m � 12.9, n � 18, q � 20.5; m�N

8. m � 23, n � 30.1, q � 42; m�Q

Determine whether the Law of Sines or the Law of Cosines should be used first tosolve �ABC. Then sole each triangle. Round angle measures to the nearest degreeand side measure to the nearest tenth.

9. a � 13, b � 18, c � 19 10. a � 6, b � 19, m�C � 38

11. a � 17, b � 22, m�B � 49 12. a � 15.5, b � 18, m�C � 72

Solve each �FGH described below. Round measures to the nearest tenth.

13. m�F � 54, f � 12.5, g � 11

14. f �20, g � 23, m�H � 47

15. f � 15.8, g � 11, h � 14

16. f � 36, h � 30, m�G � 54

17. REAL ESTATE The Esposito family purchased a triangular plot of land on which theyplan to build a barn and corral. The lengths of the sides of the plot are 320 feet, 286 feet,and 305 feet. What are the measures of the angles formed on each side of the property?

Practice The Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7

Reading to Learn MathematicsThe Law of Cosines

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7


Less

on

7-7

Pre-Activity How are triangles used in building design?


What could be a disadvantage of a triangular room?

Reading the Lesson1. Refer to the figure. According to the Law of Cosines, which

statements are correct for �DEF?

A. d2 � e2 � f 2 � ef cos D B. e2 � d2 � f 2 � 2df cos E

C. d2 � e2 � f 2 � 2ef cos D D. f 2 � d2 � e2 � 2ef cos F

E. f2 � d2 � e2 � 2de cos F F. d2 � e2 � f 2

G. �sin

dD

� � �sin

eE

� � �sin

fF

� H. d � �e2 � f�2 � 2e�f cos �D�

2. Each of the following describes three given parts of a triangle. In each case, indicatewhether you would use the Law of Sines or the Law of Cosines first in solving a trianglewith those given parts. (In some cases, only one of the two laws would be used in solvingthe triangle.)

a. SSS b. ASA

c. AAS d. SAS

e. SSA

3. Indicate whether each statement is true or false. If the statement is false, explain why.

a. The Law of Cosines applies to right triangles.

b. The Pythagorean Theorem applies to acute triangles.

c. The Law of Cosines is used to find the third side of a triangle when you are given themeasures of two sides and the nonincluded angle.

d. The Law of Cosines can be used to solve a triangle in which the measures of the threesides are 5 centimeters, 8 centimeters, and 15 centimeters.

Helping You Remember4. A good way to remember a new mathematical formula is to relate it to one you already

know. The Law of Cosines looks somewhat like the Pythagorean Theorem. Both formulasmust be true for a right triangle. How can that be?

D

dE

e

F

f


Spherical TrianglesSpherical trigonometry is an extension of plane trigonometry.Figures are drawn on the surface of a sphere. Arcs of great circles correspond to line segments in the plane. The arcs of three great circles intersecting on a sphere form a spherical triangle. Angles have the same measure as the tangent lines drawn to each great circle at the vertex. Since the sides are arcs, they too can be measured in degrees.

Solve the spherical triangle given a � 72�,b � 105�, and c � 61�.

Use the Law of Cosines.

0.3090 � (–0.2588)(0.4848) � (0.9659)(0.8746) cos Acos A � 0.5143

A � 59°

�0.2588 � (0.3090)(0.4848) � (0.9511)(0.8746) cos Bcos B � �0.4912

B � 119°

0.4848 � (0.3090)(–0.2588) � (0.9511)(0.9659) cos Ccos C � 0.6148

C � 52°

Check by using the Law of Sines.

�ssiinn

75

29

°°� � �

ssiinn

11

01

59

°°� � �

ssiinn

65

12

°°� � 1.1

Solve each spherical triangle.

1. a � 56°, b � 53°, c � 94° 2. a � 110°, b � 33°, c � 97°

3. a � 76°, b � 110°, C � 49° 4. b � 94°, c � 55°, A � 48°

A

C

B

c

ba

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

7-77-7

The sum of the sides of a spherical triangle is less than 360°.The sum of the angles is greater than 180° and less than 540°.The Law of Sines for spherical triangles is as follows.

�ssiinn

Aa

� � �ssiinn

Bb

� � �ssiinn

Cc

�

There is also a Law of Cosines for spherical triangles.

cos a � cos b cos c � sin b sin c cos A

cos b � cos a cos c � sin a sin c cos B

cos c � cos a cos b � sin a sin b cos C

ExampleExample

Chapter 7 Test, Form 177


Ass

essm

ents

Write the letter for the correct answer in the blank at the right of each question.

1. Find the geometric mean between 20 and 5.A. 100 B. 50 C. 12.5 D. 10

2. Find x in �ABC.A. 8 B. 10C. �20� D. 64

3. Find x in �PQR.A. 13 B. 15C. 16 D. �60�

4. Find x in �STU.A. 2 B. 8C. �32� D. �514�

5. Which set of measures could represent the sides of a right triangle?A. 2, 3, 4 B. 7, 11, 14C. 8, 10, 12 D. 9, 12, 15

6. Find x in �DEF.A. 6 B. 6�2�C. 6�3� D. 12

7. Find y in �XYZ.A. 7.5�3� B. 15�3�C. 15 D. 30

8. The length of the sides of a square is 10 meters. Find the length of thediagonal of the square.A. 10 m B. 10�2� mC. 10�3� m D. 20 m

9. Find x in �HJK.A. 5�2� B. 5�3�C. 10 D. 15

10. Find x in �ABC.A. 25 B. 25�2�C. 25�3� D. 100

A C

B

60� 30�50

x

H J

K60�

30�

5x

X Y

Z

15��245�

45�

y

D E

F

6

6x

S T

U

15

17x

P Q

R

12

5 x

A B

C 4

16x

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 7 Test, Form 1 (continued)77

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

11. In �QRS, �R is a right angle. Which is the ratio for the tangent of �S?

A. B.

C. D.

12. Find cos A in �ABC.

A. �274� B. �2

75�

C. �22

54� D. �

22

45�

13. Find x to the nearest tenth.A. 7.3 B. 17.3C. 18.4 D. 47.1

14. Find the angle of elevation of the sun when a pole 25 feet tall casts a shadow42 feet long.A. 30.8° B. 36.5° C. 53.5° D. 59.2°

15. Which is the angle of depression in the figure at the right?A. �AOT B. �AOBC. �TOB D. �BTO

16. Find y in �XYZ to the nearest tenth if m�Y � 36, m�X � 49, and x � 12.A. 0.04 B. 9.35 C. 14.80 D. 15.41

17. To find the distance between two points A and B on opposite sides of a river, a surveyor measures the distance from A to C as 200 feet, m�A � 72, and m�B � 37. Find the distance from A to B. Round your answer to the nearest tenth.A. 77.4 ft B. 201.2 ft C. 250.4 ft D. 314.2 ft

18. In �ABC, a � 12, b � 8, and m�A � 40. Find m�B to the nearest tenth.A. 25.4 B. 56.3 C. 59.3 D. 74.6

19. Find the third side of a triangular garden if two sides are 8 feet and 12 feetand the included angle has a measure 50.A. 7.8 ft B. 9.2 ft C. 14.4 ft D. 146.3 ft

20. In �DEF, d � 20, e � 25, and f � 30. Find m�F to the nearest tenth.A. 82.8 B. 75.5 C. 55.8 D. 47.2

Bonus In �ABC, a � 50, b � 48, and c � 40. Find m�A to the nearest tenth.

A B

C

AO

B T

67� 20

x

C A

B21

72

75

measure of leg opposite �S��measure of leg adjacent to �S

measure of leg opposite �S��measure of hypotenuse

measure of hypotenuse��measure of leg opposite �S

measure of leg adjacent to �S��measure of hypotenuse

B:

NAME DATE PERIOD

Chapter 7 Test, Form 2A77


Ass

essm

ents


1. Find the geometric mean between 7 and 12.A. 5 B. 9.5C. �19� D. �84�

2. In �PQR, RS � 4 and QS � 6. Find PS.A. 2 B. 5C. �10� D. �24�

3. Find x.A. �18� B. �14�C. 4.5 D. 3

4. Find y.A. 12 B. 11C. 9 D. 2

5. Find the length of the hypotenuse of a right triangle whose legs measure 5 and 7.A. 12 B. �24�C. �35� D. �74�

6. Find x.A. 3 B. 4C. 4�3� D. 2�5�

7. Which of the following could represent sides of a right triangle?A. 9, 40, 41 B. 8, 30, 31C. 7, 8, 15 D. �2�, �3�, �6�

8. Find c.A. 7 B. 7�2�C. 7�3� D. 14

9. Find the perimeter of a square to the nearest tenth if the length of itsdiagonal is 12 inches.A. 8.5 in. B. 33.9 in.C. 48 in. D. 67.9 in.

10. Find x.A. 4 B. 4�2�C. 4�3� D. 8�3�

x 88

60�

45�

7c

x66

8

3

y 6

2 7

x

P Q

SR 4

6

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 7 Test, Form 2A (continued)77

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

11. Find x to the nearest tenth.A. 5.8 B. 5.9C. 8.1 D. 17.3

12. In right triangle ABC, a � 12, b � 9, and c � 15. Find tan �B.

A. �43� B. �

54� C. �

34� D. �

35�

13. Find x to the nearest tenth of a degree.A. 56.3 B. 45C. 33.7 D. 29.1

14. If a 20-foot ladder makes a 65° angle with the ground, how many feet up awall will it reach? Round your answer to the nearest tenth.A. 8.5 ft B. 10 ft C. 18.1 ft D. 42.9 ft

15. A ship’s sonar finds that the angle of depression to a wreck on the bottom ofthe ocean is 12.5°. If a point on the ocean floor is 60 meters directly below theship, how many meters, to the nearest tenth, is it from that point on theocean floor to the wreck?A. 277.2 m B. 270.6 m C. 61.5 m D. 13.3 m

16. To the nearest tenth of a degree, find the angle of elevation of the sun if abuilding 100 feet tall casts a shadow 150 feet long.A. 60° B. 48.2° C. 41.8° D. 33.7°

17. When the sun’s angle of elevation is 73°, a tree tilted at an angle of 5° from the vertical casts a 20-foot shadow on the ground. Find the length of the tree to the nearest tenth.A. 6.3 ft B. 19.2 ftC. 51.1 ft D. 219.4 ft

18. In �CDE, m�C � 52, m�D � 17, and e � 28.6. Find c to the nearest tenth.A. 77.1 B. 49.1 C. 24.1 D. 18.4

19. In �PQR, p � 56, r � 17, and m�Q � 110. Find q to the nearest tenth.A. 4076.2 B. 63.8 C. 52.6 D. 3.1

20. Pete is building a kite using the dimensions given in the figure at the right. Find the measure of the angle the 2-foot edge makes with the 3-foot edge.A. 104.5 B. 85.2C. 60 D. 14.5

Bonus From a window 20 feet above the ground, the angle of elevation to the top of another building is 35°. The distance between the buildings is 52 feet. Find the height of the building to the nearest tenth of a foot.

2 ft 2 ft

3 ft3 ft4 ft

73�

5�

20-foot shadow

✹

95

x

10

36�

x

B:

NAME DATE PERIOD

Chapter 7 Test, Form 2B77


Ass

essm

ents


1. Find the geometric mean between 9 and 11.A. �99� B. �20�C. 10 D. 2

2. In �PQR, RS � 5 and QS � 8. Find PS.A. 3 B. 6.5C. �13� D. �40�

3. Find x.A. 5.5 B. �11�C. �24� D. �33�

4. Find y.A. 4 B. 5C. 8 D. 9

5. Find the length of the hypotenuse of a right triangle whose legs measure 6 and 5.A. 11 B. �11�C. �30� D. �61�

6. Find x.A. �39� B. 6C. 5�3� D. 5

7. Which of the following could represent sides of a right triangle?

A. �34�, 1, �

54� B. �3�, �5�, �15�

C. 7, 17, 24 D. 8, 15, 16

8. Find c.A. 18 B. 9�3�C. 9�2� D. 9

9. Find the perimeter of a square to the nearest tenth if the length of itsdiagonal is 16 millimeters.A. 11.3 mm B. 45.3 mmC. 90.5 mm D. 128.0 mm

10. Find x.A. 6 B. 6�2�C. 6�3� D. 12�3�

x 1212

60�

45�

9c

x8

8

10

y

4

6

x

83

P Q

SR 5

8

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Chapter 7 Test, Form 2B (continued)77

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

11. Find x.A. 8.0 B. 8.9C. 10.4 D. 10.8

12. In right triangle ABC, a � 14, b � 48, and c � 50. Find tan �A.

A. �274� B. �2

75� C. �

22

45� D. �

274�

13. Find x to the nearest tenth of a degree.A. 56.9 B. 54.5C. 33.1 D. 28.6

14. If a 24-foot ladder makes a 58° angle with the ground, how many feet up awall will it reach? Round your answer to the nearest tenth.A. 38.4 ft B. 20.8 ft C. 20.4 ft D. 12.7 ft

15. A ship’s sonar finds that the angle of depression to a wreck on the bottom ofthe ocean is 13.2°. If a point on the ocean floor is 75 meters directly below theship, how many meters, to the nearest tenth, is it from that point on theocean floor to the wreck?A. 328.4 m B. 319.8 m C. 77.0 m D. 17.6 m

16. To the nearest tenth of a degree, find the angle of elevation of the sun if abuilding 125 feet tall casts a shadow 196 feet long.A. 63.8° B. 50.4° C. 39.6° D. 32.5°

17. When the sun’s angle of elevation is 76°, a tree tilted at an angle of 4° from the vertical casts a 18-foot shadow on the ground. Find the length of the tree, to the nearest tenth.A. 250.4 ft B. 56.5 ftC. 17.7 ft D. 4.6 ft

18. In �ABC, m�A � 46, m�B � 105, and c � 19.8. Find a to the nearest tenth.A. 29.4 B. 28.5 C. 15.7 D. 14.7

19. In �LMN, l � 42, m � 61, and m�N � 108. Find n to the nearest tenth.A. 7068.4 B. 84.1 C. 79.2 D. 24.7

20. Josephine is planning a triangular garden. If the lengths of the sides are 50 feet, 80 feet, and 100 feet, what is the measure of the largest angle?A. 7.9° B. 82.1° C. 89.9° D. 97.9°

Bonus From a window 24 feet above the ground, the angle of elevation to the top of another building is 38°. The distance between the buildings is 63 feet. Find the height of the building to the nearest tenth of a foot.

76�

4�

18-foot shadow

✹

x

116

x

12

42�

B:

NAME DATE PERIOD

Chapter 7 Test, Form 2C77


Ass

essm

ents

1. Find the geometric mean between 2�5� and 5�2�.

For Questions 2–5, find x.

2. 3.

4. 5.

6. Determine whether �ABC is a right triangle. Explain your answer.

7. Find x.

8. In parallelogram ABCD, AD � 4 and m�D � 60. Find AF.

9. Find x and y.

10. Find x to the nearest tenth.

11. An A-frame house is 40 feet high and 30 feet wide. Find the measure of theangle, to the nearest tenth of a degree,that the roof makes with the floor.

12. A 30-foot tree casts a 12-foot shadow. Find the angle ofelevation of the sun to the nearest tenth of a degree.

30 ftx

40 ft

9.218�

x

4��3

60�

30�

x

y

A D

CF

B

22

x

x

y

O

C

A

B

x 2020

20

60

20x

2 12

x

6

4x

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 7 Test, Form 2C (continued)77

13. A boat is 1000 meters from a cliff. If the angle of depression from the top of the cliff to the boat is 15°, how tall is the cliff? Round your answer to the nearest tenth.

14. A plane flying at an altitude of 10,000 feet begins descendingwhen the end of the runway is below a point 50,000 feet away.Find the angle of descent (depression) to the nearest tenth of adegree.


16. Find x to the nearest tenth of a degree.

17. A tree grew at a 3° slant from the vertical. At a point 50 feet from the tree, the angle of elevation to the top of the tree is 17°. Find the length of the tree to the nearest tenth of a foot.


19. In �XYZ, m�X � 152, y � 15, and z � 19. Find x to thenearest tenth.

20. To approximate the length of a pond, a surveyor walks 400 meters from point L to point K, then turns and walks 220 meters from point Kto point E. If m �LKE � 110, find the length LE of the pond to the nearest tenth of a meter.

Bonus Find x.��6

5x

L

400 m 220 m110�

E

K

11

75x

17� 93�

50 ft

x

157

23� x

52�

26

37�x

1000 m

NAME DATE PERIOD

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 7 Test, Form 2D77


Ass

essm

ents

1. Find the geometric mean between 3�6� and 5�6�.

For Questions 2–5, find x.

2. 3.

4. 5.

6. Determine whether �ABC is a right triangle. Explain your answer.

7. Find x.

8. In parallelogram ABCD, AD � 14 and m�D � 60. Find AF.

9. Find x and y.


11. An A-frame house is 45 feet high and 32 feet wide. Find the measure of theangle, to the nearest tenth of a degree,that the roof makes with the floor.

12. A 38-foot tree casts a 16-foot shadow. Find the angle ofelevation of the sun to the nearest tenth of a degree.

32 ftx

45 ft

16�8.3

x

8��330�

60�

x

y

A D

CF

B

30

x

x

y

O

C

A

B

x 2424

24

80

30x

103

x

12

8x

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 7 Test, Form 2D (continued)77

13. A boat is 2000 meters from a cliff. If the angle of depression from the top of the cliff to the boat is 10°, how tall is the cliff? Round your answer to the nearest tenth.

14. A plane flying at an altitude of 10,000 feet begins descendingwhen the end of the runway is below a point 60,000 feet away.Find the angle of descent (depression) to the nearest tenth of adegree.



17. A tree grew at a 3° slant from the vertical. At a point 60 feet from the tree, the angle of elevation to the top of the tree is 14°. Find the length of the tree to the nearest tenth of a foot.


19. In �XYZ, m�X � 156, y � 18, and z � 21. Find x to thenearest tenth.

20. To approximate the length of a pond,a surveyor walks 420 meters from point L to point K, then turns and walks 280 meters from point K to point E. If m�LKE � 125, find the length LE of the pond to the nearest tenth of a meter.

Bonus Find x.

2��15

16x

L

420 m 280 m125�

E

K

8 9

16x

14� 93�

60 ft

x

38�

17

6

x

68�

42�

23 x

2000 m

NAME DATE PERIOD

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 7 Test, Form 377


Ass

essm

ents

1. Find the geometric mean between �29� and �

39�.

2. Find x in �PQR.

3. Find x in �XYZ.

4. If the length of one leg of a right triangle is three times thelength of the other and the hypotenuse is 20, find the length ofthe shorter leg.

5. Find the length of the altitude to the hypotenuse of a righttriangle with legs of length 3 and 4.

6. Find x.

7. Richmond is 200 kilometers due east of Teratown and Hamiltonis 150 kilometers directly north of Teratown. Find the shortestdistance in kilometers between Hamilton and Richmond.

8. Is 48, 55, 73 a Pythagorean triple? Show why or why not.

9. Find the perimeter of this square.

10. Find the perimeter of rectangle ABCD.

11. Find x and y.

12. �ABC is a 30°-60°-90° triangle with right angle A and withA�C� as the longer leg. Find the coordinates of C if A(�4, �2)and B(�4, 6).

1530�

60�x

y

12

60�

A B

CD

36��

17

89 3.5

x

x � 4 xXW Z

Y

21��

65

2x

Q

P

R

S

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

NAME DATE PERIOD

SCORE


Chapter 7 Test, Form 3 (continued)77

13. If A�B� || C�D�, find x and the length of C�D�.

14. The angle of elevation from a point on the street to the top of abuilding is 29°. The angle of elevation from another point onthe street, 50 feet farther away from the building, to the top ofthe building is 25°. To the nearest foot, how tall is the building?

15. The angle of depression from the top of a flagpole on top of alighthouse to a boat on the ocean is 37°, while the angle ofdepression from the bottom of the flagpole to the boat is 36.8°.If the boat is 1 mile away from shore and the lighthouse isright on the edge of the shore, how tall is the flagpole? Roundyour answer to the nearest foot.

16. In �JKL, m�J � 26.8, m�K � 19, and k � 17. Find � to thenearest tenth.

17. Solve �PQR for r � 22, p � 51, and m�Q � 96. Round answersto the nearest tenth.

18. Don hit a golf ball 250 yards toward the hole. However, due tothe wind, his drive is 5° off course. If the angle between the holeand where the ball lands is 97°, how far is it from where Donhit the ball to the hole? Round to the nearest tenth of a yard.

19. In �HJK, m�H � 32, k � 8, and h � 7. Find m�K. Roundyour answer(s) to the nearest tenth of a degree.

20. The distance from Albany to Bethany is 75 miles and fromBethany to Celina 105 miles. If the roads from Bethany toAlbany and from Bethany to Celina make an 87° angle, what isthe distance from Albany to Celina? Round to the nearest tenth.

Bonus A 50-foot vertical pole that stands on a hillside makes anangle of 10° with the horizontal. Two guy wires extendfrom the top of the pole to points on the hill 60 feet uphilland downhill from its base. Find the length of each guywire to the nearest tenth.

12

10

60�45�x

A B

C D

NAME DATE PERIOD

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 7 Open-Ended Assessment77


Ass

essm

ents

Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.

1. If the geometric mean between 10 and x is 6, what is x? Show how youobtained your answer.

2.

a. Max used the following equations to find x in �PQR. Is Max correct?Why or why not?

�2x� � �8

x�

x2 � 2 � 8x2 � 16x � 4

b. For �PRQ to be a right angle, what would the measure of P�S� have to be?

c. Is �PRS a 45°-45°-90° triangle? How do you know?

3. To solve for x in a triangle, when would you use sin and when would youuse sin�1? Give an example for each type of situation.

4. Draw a diagram showing where the angles of elevation and depressionare. How are the measures of these angles related?

5. Draw an obtuse triangle and label the vertices and the measures of twoangles and the length of one side. Explain how to solve the triangle.

6. Irina is solving �ABC. She plans to first use the Law of Sines to find twoof the angles. Is Irina’s plan a good one? Why or why not?

4

15

12

A C

B

8 260�

x

P Q

R

S

NAME DATE PERIOD

SCORE


Chapter 7 Vocabulary Test/Review77

Choose from the terms above to complete each sentence.

1. The square root of the product of two numbers is the of the numbers.

2. A group of three whole numbers that satisfy the equation a2 � b2 � c2, where c is the greatest number, is called a(n)

.

3. A ratio of the lengths of two sides of a right triangle is calleda(n) .

4. An angle between the line of sight and the horizontal when anobserver looks upward is called a(n) .

5. An angle between the line of sight and the horizontal when anobserver looks downward is called a(n) .

6. Three commonly used trigonometric ratios are the ,, and .

7. For �ABC, the says �sina

A� � �

sinb

B� � �

sinc

C�.

8. For �ABC, the says a2 � b2 � c2 � 2bc cos A.

9. The reciprocal of the sine is called the .

10. The reciprocal of the cosine is called the .

Define each term.

11. solving a triangle

12. Pythagorean Theorem

?

?

?

?

???

?

?

?

?

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

ambiguous caseangle of depressionangle of elevationcosecantcosine

geometric meanLaw of CosinesLaw of SinesPythagorean identityPythagorean triple

reciprocal identitiessecantsinesolving a triangle

tangenttrigonometric identitytrigonometric ratiotrigonometry

NAME DATE PERIOD

SCORE

Chapter 7 Quiz (Lessons 7–1 and 7–2)

77


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

1. Find the geometric mean between 12 and 16.

For Questions 2 and 3, find x and y.

2. 3.

4. Find x.

5. Do 19, 15, and 13 form a Pythagorean triple? Why or why not?

4

11

x

8

9

xy

5 12

xy


77

1.

2.

3.

4.

5.

6.

7.

8.

9.

For Questions 1 and 2, find x.

1. 2.

For Questions 3 and 4, find x to the nearest tenth.

3. 4.

5. A rectangle has a diagonal 20 inches long that forms angles of60° and 30° with the sides. Find the perimeter of the rectangle.

6. Find sin 52°. Round to the nearest ten-thousandth.

7. If cos A � 0.8945, find �A to the nearest tenth of a degree.

8. The distance along a hill is 24 feet. If the land slopes uphill atan angle of 8°, find the vertical distance from the top to thebottom of the hill.

9. A surveyor is standing on horizontal ground level with thebase of a skyscraper. The angle formed by the line segmentfrom his position to the top of the skyscraper is 31°. Theheight of the building is 1200 feet. Find the distance from thebuilding to the surveyor to the nearest foot.

1731�

x

13

11

x

60� 30�

6x

45�

6x

NAME DATE PERIOD

SCORE



77

1.

2.

3.

4.

5.

1. Name the angle of elevation in the figure.


3. Solve �ABC. Round your answers to the nearest tenth.

4. A triangular lot has 500 feet of frontage along a river. Theother two sides make angles of 48° and 75° with the riverfrontside. Find the length of the shortest side to the nearest foot.

5. STANDARDIZED TEST PRACTICE A squirrel 37 feet up in atree sees a dog 29 feet from the base of the tree. Find themeasure of the angle of depression to the nearest tenth of adegree.A. 38.4 B. 51.9 C. 45.0 D. 128.1

49�

11 18

A C

B

22�76�

10x

P

S

Q

R

NAME DATE PERIOD

SCORE

Chapter 7 Quiz (Lesson 7–7)

77

1.

2.

3.

4.

5.

For Questions 1 and 2, find x to the nearest tenth.

1. 2.

3. Solve �RST. Round your answers to the nearest degree.

4. A hiker is 6 miles from a tower and 8 miles from the lodge.She estimates that the angle between her path to the towerand her path to the lodge is 42°. Find the distance from thetower to the lodge to the nearest tenth of a mile.

5. STANDARDIZED TEST PRACTICE For �ABC, find a to thenearest tenth if m�A � 96, b � 41, and c � 50.A. 66.3 B. 67.9 C. 4395.3 D. 4609.6

48 61

76T S

R

6x

18

15

82�

x

32

23

NAME DATE PERIOD

SCORE

Chapter 7 Mid-Chapter Test (Lessons 7–1 through 7–4)

77


Ass

essm

ents

1. Find the geometric mean between 7 and 9.A. �63� B. 16 C. 8 D. 2

2. Find x.

A. �216� B.

C. 6�55� D.

3. Find sin C.

A. �2� B.

C. D.

4. Find x to the nearest tenth.A. 14 B. 18.4C. 21.1 D. 32.2

5. Find y to the nearest tenth of a degree.A. 144.9 B. 60.0C. 44.7 D. 35.1

27

19y

28

49�x

�23��5

�23��2�

�2��5

5��2

��23B C

A

�23��5

�2��5

924

24

x

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

Part II

For Questions 6–8, find x and y.

6. 7.

8.

9. Do 56, 90, 106 form a Pythagorean triple? Why or why not?

10. Guy wires 80 feet long support a 65-foot tall telephone pole. Tothe nearest tenth of a degree, what angle will the wires makewith the ground?

30�

60�

45�

8��3

yx

60� 45�

20 yx

3

6 y

x

Part I Write the letter for the correct answer in the blank at the right of each question.


Chapter 7 Cumulative Review(Chapters 1–7)

77

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1. Name the vertex and sides, then classify �IFT.(Lesson 1-4)

For Questions 2 and 3, complete the following proof. (Lesson 2-7)

Given: J�K� � L�M�H�J� � K�L�

Prove: H�K� � K�M�Proof:Statements Reasons1. J�K� � L�M�, H�J� � K�L� 1. Given2. JK � LM, HJ � KL 2.3. 3. Segment Addition Post.4. HJ � JK � KL � LM 4. Substitution Prop.5. HK � KM 5. Substitution Prop.6. H�K� � K�M� 6. Def. of � segments

For Questions 4 and 5, use the figure at the right.

4. Find the measure of the numbered angles if m�ABC � 57 and m�BCE � 98. (Lesson 4-2)

5. If B�D� is a median, AD � 2x � 6, and DC � 22.5 � 4x, find AC. (Lesson 5-1)

6. Write an inequality to describe the possible values of x. (Lesson 5-5)

7. A band of sequins that measures 108 inches is cut into twopieces so that their lengths are in a 5:7 ratio. Find the lengthof each piece. (Lesson 6-1)

8. Stan invests $1875 in a certificate of deposit that earns 4.5%interest compounded annually. Find the balance of his accountafter 4 years. (Lesson 6-6)


9. Find QP to the nearest tenth.(Lesson 7-2)

10. Find LM and PM. (Lesson 7-3)

60�30�10

9

P

ML

Q

R

5

85�117�

1212

145x � 6

E

CB

AD

25�1

23

4

5

41�

(Question 3)(Question 2)

H

J

K L M

F

TI 89�

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–7)


1. If T�A� bisects �YTB, T�C� bisects �BTZ,m�YTA � 4y � 6, and m�BTC � 7y � 4,find m�CTZ. (Lesson 1-4)

A. 52 B. 38C. 25 D. 8

2. Which statement is always true? (Lesson 2-5)

E. If right �QPR has sides q, p, and r, where r is the hypotenuse,then r2 � p2 � q2.

F. If E�F� || H�J�, then EF � HJ.G. If lines KL and VT are cut by a transversal, then K�L� || V�T�.H. If D�R� and R�H� are congruent, then R bisects D�H�.

3. The equation for PT�� is y � 2 � 8(x � 3). Determine an equationfor a line perpendicular to PT��. (Lesson 3-4)

A. y � �18�x � 7 B. y � 8x � 13

C. y � ��18�x � 2 D. y � �8x

4. Angle Y in �XYZ measures 90°. X�Y� and Y�Z� each measure 16meters. Classify �XYZ. (Lesson 4-1)

E. acute and isoscelesF. equiangular and equilateralG. right and scaleneH. right and isosceles

5. Two sides of a triangle measure 4 inches and 9 inches. Determinewhich cannot be the perimeter of the triangle. (Lesson 5-4)

A. 19 in. B. 21 in. C. 23 in. D. 26 in.

6. �ABC � �STR, so �AC

BA� � . (Lesson 6-2)

E. �BAB

C� F. �RST

S� G. �TR

RS� H. �

RST

S�

7. The Petronas Towers in Kuala Lumpur, Malaysia, are 452 meterstall. A woman who is 1.75 meters tall stands 120 meters from thebase of one tower. Find the angle of elevation between the woman’shat and the top of the tower to the nearest tenth. (Lesson 7-5)

A. 14.8° B. 15.4° C. 74.5° D. 75.1°

8. Which equation can be used to find x? (Lesson 7-4)

E. x � y sin 73° F. x � y cos 73°G. x � �cos

y73°� H. x � �sin

y73°�

73�

x

y

?

Y ZT

A CB

NAME DATE PERIOD

SCORE 77

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

1.

2.

3.

4.

5.

6.

7.

8. E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

E F G H

A B C D

Ass

essm

ents


Standardized Test Practice (continued)

9. Find the measure of the smaller of twocomplementary angles whose measures differby 23. (Lesson 1-5)

10. How many counterexamples are necessary toprove that a statement is false? (Lesson 2-3)

11. Find x so that � || m . (Lesson 3-5)

12. Find c to the nearest tenth. (Lesson 7-6)

26.1�

27.7�

126.2�

19

33

A

B

C

c

134 � 4x6x � 17

m�

NAME DATE PERIOD

77

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

Part 3: Short Response

Instructions: Show your work or explain in words how you found your answer.

9. 10.

11. 12.

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.

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13. If �DEF � �HJK, m�D � 26, m�J � 3x � 5, and m�F � 92, find x. (Lesson 4-3)

14. Use the Exterior Angle Inequality Theorem to list all of the angles whose measures are less than m�1. (Lesson 5-2)


15. Determine whether �EFH � �JGH. (Lesson 6-3)

16. If G is the midpoint of F�H�,find x. (Lesson 7-3)

60�18

30�E JH

F

Gx

1 2 4

3 56 7

8A DC

B

13.

14.

15.

16.

3 3 . 5 1

1 4 . 5 1 8 . 0

Unit 2 Review(Chapter 4–7)

77


Ass

essm

ents

1. Use a protractor to classify �UVW, �UWX,and �XWY as acute, equiangular, obtuse,or right.

2. In the figure, �1 � �2. Find the measures of the numbered angles.

3. Name the corresponding congruent sides for �AFP � �STX.

4. Determine whether �ABC � �PQR given A(2, �7), B(5, 3),C(�4, 6), P(8, �1), Q(11, 9), and R(2, 12).

5. In the figure, L�K� bisects �JKM and �KLJ � �KLM. Determine which theorem or postulate can be used to prove that �JKL � �MKL.

6. Triangle ABC is isosceles with AB � BC. Name a pair ofcongruent angles in this triangle.

7. Name the missing coordinates for isosceles right �JKL with legs b units long.

For Questions 8 and 9, refer to the figure.

8. Find a and m�ZWT if Z�W� is an altitude of �XYZ, m�ZWT � 3a � 5, and m�TWY � 5a � 13.

9. Determine which angle has the greatest measure: �YWZ,�WZY, or �ZYW.

10. Mr. Ramirez bought a stove and a dishwasher for just over$1206. State the assumption you would make to start anindirect proof to show that at least one of the appliances costmore than $603.

X W Y

T

Z

x

y

L(?, ?)

K(?, ?)

J(?, ?)

K L

J

M

65�

70�110�13

2

D

E F H

G

V

U

W

X

Y1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

NAME DATE PERIOD

SCORE


Unit 2 Review (continued)77

11. Determine whether 128 feet, 136 feet, and 245 feet can be thelengths of the sides of a triangle.

12. Casey has a 13-inch television and a 52-inch television in herhome. What is the ratio of the sizes of the smaller and largerTVs?

13. If �EFG � �EJK, find x, JK,KG, and the scale factor relating �EFG to �EJK.

14. Find y.

15. Find the perimeter of �ABCif �ABC � �XYZ.

16. Alex has $750 in a bank account that earns 2.7% interest. Ifthe interest is compounded annually and he does not makeany withdrawals, find the balance of his account after 3 years.

17. Find the geometric mean between 27 and 42 to the nearesttenth.

18. Determine whether 27, 120, and 123 are the measures of thesides of a right triangle. Then state whether they form aPythagorean triple.

19. The diagonal of a square is 56 centimeters long. Find theperimeter of the square to the nearest tenth.

20. Find m�P to the nearest tenth in right �MNP for M(3, 6),N(3, �8), and P(�5, �8).

For Questions 21 and 22, refer to the figure.

21. Find m�S if m�T � 68, t � 65, and s � 33.

22. Solve �RST if t � 17, s � 11, and m�R � 40.

S

TR

r

s

t

X Z

Y

AC

B6

46

3028

y � 4

2y � 1

7

7

32�12

10

189

E

F

GK

J(x � 7)�

NAME DATE PERIOD

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

First Semester Test(Chapter 1–7)

77


Ass

essm

ents

For Questions 1–7, write the letter for the correct answer in the blank atthe right of each question.

1. Angles AFH and HFB form a linear pair and m�AFH � 83. Find m�HFB.A. 164 B. 97 C. 83 D. 41.5

2. Given C(2, 5), D(7, 0), and F(13, �6), which of the following is a trueconjecture?A. �CDF is a right triangle. B. �CDF is an isosceles triangle.C. �CDF is an equilateral triangle. D. C, D, and F do not form a triangle.

3. Which is the inverse of the statement If x � 5, then x � 3 � 8?A. If x � 3 � 8, then x � 5. B. If x � 5, then x � 3 � 8.C. If x � 5, then x � 3 � 8. D. If x � 3 � 8, then x � 5.

4. Find the slope of a line that is perpendicular to GH��.

A. �23� B. �

32�

C. � �23� D. � �

32�

5. Find the distance between parallel lines � and m whose equations are

y � �34�x � 4 and y � �

34�x � �

94�.

A. 4 B. 5 C. 9 D. �94�

6. Find sin P.

A. �15

40� B. �

14

48�

C. �45

80� D. 1

7. Find m�G.A. 30° B. 32°C. 35° D. 55.8°

32

2218

H

GF

50

48

14

P R

Q

x

y

O

G

H

8.

9.

10.

11.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

6.

7.

8. Find c and PK if P is between L and K, LP � c � 22, PK � 5c,and LK � 34. Does P bisect L�K�?

9. Determine the distance between A(15, �12), and B(�30, 48) ona coordinate plane. State the coordinates of the midpoint of A�B�.

Justify each statement with a property or definition.

10. If A�C� � B�D�, then AC � BD.

11. If �2 and �3 are complementary, then m�2 � m�3 � 90.


First Semester Test (continued)77

12. If the measures of two angles of a triangle are 24 and 30, isthe triangle acute, obtuse, or right? Explain your reasoning.

13. Identify the congruent triangles in the figure.

For Questions 14 and 15, refer to the figure. Triangle ABC is anisosceles right triangle.

14. If C�D� bisects �C, find m�1 and m�2.

15. Determine the coordinates of A, B, and C, if the triangle haslegs n units long.

For Questions 16–18, refer to the figure.

16. Write a statement using �, �,or � to describe the measures of �DBC and �DCB.

17. Write an inequality to represent the possible measures of D�E�.

18. If m�FBC � 3x � 1 and m�CBD � 34, write an inequality todescribe the possible values of x.

19. Identify the similar triangles, find MN, and state the scale factor fromthe smaller triangle to the largertriangle.

20. Find the first three iterates of 4(x � 3) if x initially equals 0.

21. A plane is flying at 35,000 feet, and the pilot wants to descendto 22,000 feet over the next 60 miles. What should be his angleof depression to the nearest tenth? (Hint: 5280 feet � 1 mile)

22. Solve �DEF if DE � 58, EF � 62, and m�E � 49. Roundangle measures to the nearest degree and side measures tothe nearest tenth.

L

N

MK

J

5542

63

14

14

10

1213

F

B D

EC

x

y

C

B

A

D

1

2

K P

N

M

L

J

NAME DATE PERIOD

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

Standardized Test PracticeStudent Record Sheet (Use with pages 398–399 of the Student Edition.)

77

© Glencoe/McGraw-Hill A1 Glencoe Geometry

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5

3 6 DCBADCBA

DCBADCBA

DCBADCBADCBA

NAME DATE PERIOD

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 3 Open-EndedPart 3 Open-Ended

Solve the problem and write your answer in the blank.

For Questions 8, 9, 11, and 12, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

8 (grid in) 8 9

9 (grid in)

10

11 (grid in)

12 (grid in)

11 12

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Record your answers for Question 13 on the back of this paper.


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13.1

2,16

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14.1

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),H

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8.G

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yes;

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of

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e th

e m

easu

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of t

he

sid

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f a

righ

t tr

ian

gle.

Th

en s

tate

wh

eth

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hey

for

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hag

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n t

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le.

11.9

,40,

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yes,

yes

no

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yes,

yes

14.�

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yes,

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yes,

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17.C

ON

STR

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he

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of a

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a w

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th

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11

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lon

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7-2

7-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csT

he

Pyt

hag

ore

an T

heo

rem

an

d It

s C

onv

erse

NA

ME

____

____

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7-2

7-2

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Lesson 7-2

Pre-

Act

ivit

yH

ow a

re r

igh

t tr

ian

gles

use

d t

o b

uil

d s

usp

ensi

on b

rid

ges?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-2

at

the

top

of p

age

350

in y

our

text

book

.

Do

the

two

righ

t tr

ian

gles

sh

own

in

th

e dr

awin

g ap

pear

to

be s

imil

ar?

Exp

lain

you

r re

ason

ing.

Sam

ple

an

swer

:N

o;

thei

r si

des

are

no

tp

rop

ort

ion

al.I

n t

he

smal

ler

tria

ng

le,t

he

lon

ger

leg

is m

ore

th

antw

ice

the

len

gth

of

the

sho

rter

leg

,wh

ile in

th

e la

rger

tri

ang

le,

the

lon

ger

leg

is le

ss t

han

tw

ice

the

len

gth

of

the

sho

rter

leg

.

Rea

din

g t

he

Less

on

1.E

xpla

in i

n y

our

own

wor

ds t

he

diff

eren

ce b

etw

een

how

th

e P

yth

agor

ean

Th

eore

m i

s u

sed

and

how

th

e C

onve

rse

of t

he

Pyt

hag

orea

n T

heo

rem

is

use

d.S

amp

le a

nsw

er:T

he

Pyt

hag

ore

an T

heo

rem

is u

sed

to

fin

d t

he

thir

d s

ide

of

a ri

gh

t tr

ian

gle

ifyo

u k

no

w t

he

len

gth

s o

f an

y tw

o o

f th

e si

des

.Th

e co

nver

se is

use

d t

ote

ll w

het

her

a t

rian

gle

wit

h t

hre

e g

iven

sid

e le

ng

ths

is a

rig

ht

tria

ng

le.

2.R

efer

to

the

figu

re.F

or t

his

fig

ure

,wh

ich

sta

tem

ents

are

tru

e?

A.

m2

�n

2�

p2B

.n2

�m

2�

p2B

,E,F

,G

C.

m2

�n

2�

p2D

.m2

�p2

� n

2

E.

p2�

n2

� m

2F.

n2

� p

2�

m2

G.n

��

m2

��

p2 �H

.p�

�m

2�

�n

2 �

3.Is

th

e fo

llow

ing

stat

emen

t tr

ue

or f

alse

?A

Pyt

hago

rean

tri

ple

is a

ny g

roup

of

thre

e nu

mbe

rs f

or w

hich

the

sum

of

the

squa

res

of t

hesm

alle

r tw

o nu

mbe

rs is

equ

al t

o th

e sq

uare

of

the

larg

est

num

ber.

Exp

lain

you

r re

ason

ing.

Sam

ple

an

swer

:Th

e st

atem

ent

is f

alse

bec

ause

in a

Pyt

hag

ore

an t

rip

le,

all t

hre

e n

um

ber

s m

ust

be

wh

ole

nu

mb

ers.

4.If

x,y

,an

d z

form

a P

yth

agor

ean

tri

ple

and

kis

a p

osit

ive

inte

ger,

wh

ich

of

the

foll

owin

ggr

oups

of

nu

mbe

rs a

re a

lso

Pyt

hag

orea

n t

ripl

es?

B,D

A.3

x,4y

,5z

B.3

x,3y

,3z

C.�

3x,�

3y,�

3zD

.kx,

ky,k

z

Hel

pin

g Y

ou

Rem

emb

er

5.M

any

stud

ents

who

stu

died

geo

met

ry lo

ng a

go r

emem

ber

the

Pyt

hago

rean

The

orem

as

the

equa

tion

a2

�b2

�c2

,but

can

not

tell

you

wha

t th

is e

quat

ion

mea

ns.A

for

mul

a is

use

less

if y

ou d

on’t

know

wh

at i

t m

ean

s an

d h

ow t

o u

se i

t.H

ow c

ould

you

hel

p so

meo

ne

wh

o h

asfo

rgot

ten

the

Pyt

hago

rean

The

orem

rem

embe

r th

e m

eani

ng o

f th

e eq

uati

on a

2�

b2�

c2?

Sam

ple

an

swer

:D

raw

a r

igh

t tr

ian

gle

.Lab

el t

he

len

gth

s o

f th

e tw

o le

gs

as a

and

ban

d t

he

len

gth

of

the

hyp

ote

nu

se a

s c.

pm

n

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Th

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u h

ave

lear

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th

at t

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mea

sure

of

the

alti

tude

fro

m t

he

vert

ex o

fth

e ri

ght

angl

e of

a r

igh

t tr

ian

gle

to i

ts h

ypot

enu

se i

s th

e ge

omet

ric

mea

n b

etw

een

th

e m

easu

res

of t

he

two

segm

ents

of

the

hyp

oten

use

.Is

th

e co

nve

rse

of t

his

th

eore

m t

rue?

In

ord

er t

o fi

nd

out,

it w

ill

hel

pto

rew

rite

th

e or

igin

al t

heo

rem

in

if-

then

for

m a

s fo

llow

s.

If �

AB

Qis

a r

igh

t tr

ian

gle

wit

h r

igh

t an

gle

at Q

,th

en

QP

is

the

geom

etri

c m

ean

bet

wee

n A

Pan

d P

B,w

her

e P

is b

etw

een

Aan

d B

and

Q �P �

is p

erpe

ndi

cula

r to

A �B �

.

1.W

rite

th

e co

nve

rse

of t

he

if-t

hen

for

m o

f th

e th

eore

m.

If Q

Pis

th

e g

eom

etri

c m

ean

bet

wee

n A

Pan

d

PB

,wh

ere

Pis

bet

wee

n A

and

Ban

d Q �

P ��

A �B �

,th

en �

AB

Qis

a r

igh

t tr

ian

gle

wit

h r

igh

t an

gle

at

Q.

2.Is

th

e co

nve

rse

of t

he

orig

inal

th

eore

m t

rue?

Ref

er

to t

he

figu

re a

t th

e ri

ght

to e

xpla

in y

our

answ

er.

Yes;

(PQ

)2�

(AP

)(P

B)

imp

lies

that

�P AQ P��

� PPQB �

.

Sin

ce b

oth

�A

PQ

and

�Q

PB

are

rig

ht

ang

les,

they

are

co

ng

ruen

t.T

her

efo

re�

AP

Q�

�Q

PB

by S

AS

sim

ilari

ty.S

o

�A

��

PQ

Ban

d �

AQ

P�

�B

.Bu

t th

e ac

ute

an

gle

s o

f �

AQ

Par

e co

mp

lem

enta

ry a

nd

m

�A

QB

�m

�A

QP

�m

�P

QB

.Hen

ce

m�

AQ

B�

90 a

nd

�A

QB

is a

rig

ht

tria

ng

le

wit

h r

igh

t an

gle

at

Q.

You

may

fin

d it

in

tere

stin

g to

exa

min

e th

e ot

her

th

eore

ms

inC

hap

ter

7 to

see

wh

eth

er t

hei

r co

nve

rses

are

tru

e or

fal

se.Y

ou w

ill

nee

d to

res

tate

th

e th

eore

ms

care

full

y in

ord

er t

o w

rite

th

eir

con

vers

es.

Q

BP

A

Q

BP

A

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-2

7-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Sp

ecia

l Rig

ht T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-3

7-3

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Lesson 7-3

Pro

per

ties

of

45°-

45°-

90°

Tria

ng

les

Th

e si

des

of a

45°

-45°

-90°

righ

t tr

ian

gle

hav

e a

spec

ial

rela

tion

ship

.

If t

he

leg

of a

45°

-45°

-90°

righ

t tr

ian

gle

is x

un

its,

show

th

at t

he

hyp

oten

use

is

x�2�

un

its.

Usi

ng

the

Pyt

hag

orea

n T

heo

rem

wit

h

a�

b�

x,th

en

c2�

a2�

b2

�x2

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2

c�

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2�

x��

x

x2

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45�

In a

45°

-45°

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righ

t tr

ian

gle

the

hyp

oten

use

is

�2�

tim

es

the

leg.

If t

he

hyp

oten

use

is

6 u

nit

s,fi

nd

th

e le

ngt

h o

f ea

ch l

eg.

Th

e h

ypot

enu

se i

s �

2�ti

mes

th

e le

g,so

divi

de t

he

len

gth

of

the

hyp

oten

use

by

�2�.

a� � � �

3 �2�

un

its

6�2�

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2�

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ple1

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Exer

cises

Fin

d x

.

1.2.

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the

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of

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0 in

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in.

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ind

the

diag

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Pro

per

ties

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90°

Tria

ng

les

Th

e si

des

of a

30°

-60°

-90°

righ

t tr

ian

gle

also

hav

e a

spec

ial

rela

tion

ship

.

In a

30°

-60°

-90°

righ

t tr

ian

gle,

show

th

at t

he

hyp

oten

use

is

twic

e th

e sh

orte

r le

g an

d t

he

lon

ger

leg

is �

3�ti

mes

th

e sh

orte

r le

g.

�M

NQ

is a

30°

-60°

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righ

t tr

ian

gle,

and

the

len

gth

of

the

hyp

oten

use

M �N�

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wo

tim

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he

len

gth

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the

shor

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sin

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e P

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eore

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a2�

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2

a�

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2�

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3�

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30°

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-90°

righ

t tr

ian

gle,

the

hyp

oten

use

is

5 ce

nti

met

ers.

Fin

d t

he

len

gth

s of

th

e ot

her

tw

o si

des

of

the

tria

ngl

e.If

th

e h

ypot

enu

se o

f a

30°-

60°-

90°

righ

t tr

ian

gle

is 5

cen

tim

eter

s,th

en t

he

len

gth

of

the

shor

ter

leg

is h

alf

of 5

or

2.5

cen

tim

eter

s.T

he

len

gth

of

the

lon

ger

leg

is �

3�ti

mes

th

e le

ngt

h o

f th

e sh

orte

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g,or

(2.

5)(�

3�)ce

nti

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ers.

Fin

d x

and

y.

1.2.

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1;0.

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32

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ind

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len

gth

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ltit

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of t

he

tria

ngl

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th

e n

eare

st t

enth

of

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met

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9.2

cm

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n a

ltit

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of

an e

quil

ater

al t

rian

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is 8

.3 m

eter

s.F

ind

the

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met

er o

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ian

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e n

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st t

enth

of

a m

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20

xy

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xy

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y

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x y30

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Stu

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)

Sp

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s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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____

____

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ER

IOD

____

_

7-3

7-3

Exer

cises

Exer

cises

Exam

ple1

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ple1

Exam

ple2

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ple2

�M

NP

is a

n eq

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tera

ltr

iang

le.

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NQ

is a

30°

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right

tria

ngle

.



An

swer

s

Skil

ls P

ract

ice

Sp

ecia

l Rig

ht T

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gle

s

NA

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7-3

7-3

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Lesson 7-3

Fin

d x

and

y.

1.2.

3.

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6�2�

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For

Exe

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–9,u

se t

he

figu

re a

t th

e ri

ght.

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a�

11,f

ind

ban

d c.

b�

11�

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8.If

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9.If

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0 an

d 1

1,u

se t

he

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re a

t th

e ri

ght.

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he

peri

met

er o

f th

e sq

uar

e is

30

inch

es.F

ind

the

len

gth

of

B�C�

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ind

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diag

onal

B�D�

.

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.or

abo

ut

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1 in

.

12.T

he

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e eq

uil

ater

al t

rian

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is 6

0 m

eter

s.F

ind

the

le

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10.T

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s.F

ind

the

len

gth

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6.5�

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d th

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for

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nd

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.

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13.B

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e of

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th

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m a

diag

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hw

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th

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7-3

7-3



Readin

g t

o L

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Math

em

ati

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pec

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s

NA

ME

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7-3

7-3

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Lesson 7-3

Pre-

Act

ivit

yH

ow i

s tr

ian

gle

tili

ng

use

d i

n w

allp

aper

des

ign

?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-3

at

the

top

of p

age

357

in y

our

text

book

.•

How

can

you

mos

t co

mpl

etel

y de

scri

be t

he

larg

er t

rian

gle

and

the

two

smal

ler

tria

ngl

es i

n t

ile

15?

Sam

ple

an

swer

:Th

e la

rger

tri

ang

le is

an is

osc

eles

ob

tuse

tri

ang

le.T

he

two

sm

alle

r tr

ian

gle

s ar

eco

ng

ruen

t sc

alen

e ri

gh

t tr

ian

gle

s.•

How

can

you

mos

t co

mpl

etel

y de

scri

be t

he

larg

er t

rian

gle

and

the

two

smal

ler

tria

ngl

es i

n t

ile

16?

(In

clu

de a

ngl

e m

easu

res

in d

escr

ibin

g al

l th

etr

iang

les.

)S

amp

le a

nsw

er:T

he

larg

er t

rian

gle

is e

qu

ilate

ral,

soea

ch o

f it

s an

gle

mea

sure

s is

60.

Th

e tw

o s

mal

ler

tria

ng

les

are

con

gru

ent

rig

ht

tria

ng

les

in w

hic

h t

he

ang

le m

easu

res

are

30,6

0,an

d 9

0.

Rea

din

g t

he

Less

on

1.S

upp

ly t

he

corr

ect

nu

mbe

r or

nu

mbe

rs t

o co

mpl

ete

each

sta

tem

ent.

a.In

a 4

5°-4

5°-9

0°tr

ian

gle,

to f

ind

the

len

gth

of

the

hyp

oten

use

,mu

ltip

ly t

he

len

gth

of

a

leg

by

.

b.

In a

30°

-60°

-90°

tria

ngl

e,to

fin

d th

e le

ngt

h o

f th

e h

ypot

enu

se,m

ult

iply

th

e le

ngt

h o

f

the

shor

ter

leg

by

.

c.In

a 3

0°-6

0°-9

0°tr

iang

le,t

he lo

nger

leg

is o

ppos

ite

the

angl

e w

ith

a m

easu

re o

f .

d.

In a

30°

-60°

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tria

ngl

e,to

fin

d th

e le

ngt

h o

f th

e lo

nge

r le

g,m

ult

iply

th

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ngt

h o

f

the

shor

ter

leg

by

.

e.In

an

iso

scel

es r

igh

t tr

ian

gle,

each

leg

is

oppo

site

an

an

gle

wit

h a

mea

sure

of

.

f.In

a 3

0°-6

0°-9

0°tr

ian

gle,

to f

ind

the

len

gth

of

the

shor

ter

leg,

divi

de t

he

len

gth

of

the

lon

ger

leg

by

.

g.In

30

°-60

°-90

°tr

ian

gle,

to f

ind

the

len

gth

of

the

lon

ger

leg,

divi

de t

he

len

gth

of

the

hyp

oten

use

by

and

mu

ltip

ly t

he

resu

lt b

y .

h.

To

fin

d th

e le

ngt

h o

f a

side

of

a sq

uar

e,di

vide

th

e le

ngt

h o

f th

e di

agon

al b

y .

2.In

dica

te w

het

her

eac

h s

tate

men

t is

alw

ays,

som

etim

es,o

r n

ever

tru

e.a.

Th

e le

ngt

hs

of t

he

thre

e si

des

of a

n i

sosc

eles

tri

angl

e sa

tisf

y th

e P

yth

agor

ean

Th

eore

m.

som

etim

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.T

he

len

gth

s of

th

e si

des

of a

30°

-60°

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tria

ngl

e fo

rm a

Pyt

hag

orea

n t

ripl

e.n

ever

c.T

he

len

gth

s of

all

th

ree

side

s of

a 3

0°-6

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0°tr

ian

gle

are

posi

tive

in

tege

rs.

nev

er

Hel

pin

g Y

ou

Rem

emb

er3.

Som

e st

ude

nts

fin

d it

eas

ier

to r

emem

ber

mat

hem

atic

al c

once

pts

in t

erm

s of

spe

cifi

cn

um

bers

rat

her

th

an v

aria

bles

.How

can

you

use

spe

cifi

c n

um

bers

to

hel

p yo

u r

emem

ber

the

rela

tion

ship

bet

wee

n t

he

len

gth

s of

th

e th

ree

side

s in

a 3

0°-6

0°-9

0°tr

ian

gle?

Sam

ple

an

swer

:D

raw

a 3

0�-6

0�-9

0�tr

ian

gle

.Lab

el t

he

len

gth

of

the

sho

rter

leg

as

1.T

hen

th

e le

ng

th o

f th

e hy

po

ten

use

is 2

,an

d t

he

len

gth

of

the

lon

ger

leg

is �

3�.Ju

st r

emem

ber

:1,

2,�

3�.

�2�

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602

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Co

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Val

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of

Sq

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oo

tsT

he

diag

ram

at

the

righ

t sh

ows

a ri

ght

isos

cele

s tr

ian

gle

wit

h

two

legs

of

len

gth

1 i

nch

.By

the

Pyt

hag

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n T

heo

rem

,th

e le

ngt

h

of t

he

hyp

oten

use

is

�2�

inch

es.B

y co

nst

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an a

djac

ent

righ

t tr

ian

gle

wit

h l

egs

of �

2�in

ches

an

d 1

inch

,you

can

cre

ate

a se

gmen

t of

len

gth

�3�.

By

con

tin

uin

g th

is p

roce

ss a

s sh

own

bel

ow,y

ou c

an c

onst

ruct

a

“wh

eel”

of s

quar

e ro

ots.

Th

is w

hee

l is

cal

led

the

“Wh

eel

of T

heo

doru

s”af

ter

a G

reek

ph

ilos

oph

er w

ho

live

d ab

out

400

B.C

.

Con

tin

ue

con

stru

ctin

g th

e w

hee

l u

nti

l yo

u m

ake

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gmen

t of

len

gth

�18�

.

��

1

1

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3�

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2

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rich

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NA

ME

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____

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ER

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7-3

7-3

1

1

11

1

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An

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e a

nd I

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rven

tion

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on

om

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NA

ME

____

____

____

____

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7-4

7-4

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Lesson 7-4

Trig

on

om

etri

c R

atio

sT

he

rati

o of

th

e le

ngt

hs

of t

wo

side

s of

a r

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t tr

ian

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is c

alle

d a

trig

onom

etri

c ra

tio.

Th

e th

ree

mos

t co

mm

on r

atio

s ar

e si

ne,

cosi

ne,

and

tan

gen

t,w

hic

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re a

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spec

tive

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sin

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Use

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ric

Rat

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In a

rig

ht

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ngl

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you

kn

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sure

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tw

o si

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f yo

u k

now

th

e m

easu

res

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ne

side

an

d an

acu

te a

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en y

ou c

an u

se t

rigo

nom

etri

cra

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to

fin

d th

e m

easu

res

of t

he

mis

sin

g si

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ian

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d x

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7-4

7-4

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7-4

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oe G

eom

etry

Lesson 7-4

Use

�R

ST

to f

ind

sin

R,c

os R

,tan

R,s

in S

,cos

S,a

nd

tan

S.

Exp

ress

eac

h r

atio

as

a fr

acti

on a

nd

as

a d

ecim

al t

o th

e n

eare

st h

un

dre

dth

.

1.r

�16

,s�

30,t

�34

2.r

�10

,s�

24,t

�26

sin

R�

�1 36 4��

0.47

;si

n R

��1 20 6�

�0.

38;

cos

R�

�3 30 4��

0.88

;co

s R

��2 24 6�

�0.

92;

tan

R�

�1 36 0��

0.53

;ta

n R

��1 20 4�

�0.

42;

sin

S�

�3 30 4��

0.88

;si

n S

��2 24 6�

�0.

92;

cos

S�

�1 36 4��

0.47

;co

s S

��1 20 6�

�0.

38;

tan

S�

�3 10 6��

1.88

tan

S�

�2 14 0��

2.4

Use

a c

alcu

lato

r to

fin

d e

ach

val

ue.

Rou

nd

to

the

nea

rest

ten

-th

ousa

nd

th.

3.si

n 5

0.08

724.

tan

23

0.42

455.

cos

610.

4848

6.si

n 7

5.8

0.96

947.

tan

17.

30.

3115

8.co

s 52

.90.

6032

Use

th

e fi

gure

to

fin

d e

ach

tri

gon

omet

ric

rati

o.E

xpre

ss

answ

ers

as a

fra

ctio

n a

nd

as

a d

ecim

al r

oun

ded

to

the

nea

rest

ten

-th

ousa

nd

th.

9.ta

n C

10.s

in A

11.c

os C

� 49 0��

0.22

50�4 40 1�

�0.

9756

�4 40 1��

0.97

56

Fin

d t

he

mea

sure

of

each

acu

te a

ngl

e to

th

e n

eare

st t

enth

of

a d

egre

e.

12.s

in B

�0.

2985

17.4

13.t

an A

�0.

4168

22.6

14.c

os R

�0.

8443

32.4

15.t

an C

�0.

3894

21.3

16.c

os B

�0.

7329

42.9

17.s

in A

�0.

1176

6.8

Fin

d x

.Rou

nd

to

the

nea

rest

ten

th.

18.

19.

20.

28.8

73.5

15.9

19

x

33�

UL

S

27

x�8

BAC

27

x�

13 BA

C

41

409

B

A

C

sR

S Trt

©G

lenc

oe/M

cGra

w-H

ill37

2G

lenc

oe G

eom

etry

Use

�L

MN

to f

ind

sin

L,c

os L

,tan

L,s

in M

,cos

M,a

nd

tan

M.

Exp

ress

eac

h r

atio

as

a fr

acti

on a

nd

as

a d

ecim

al t

o th

e n

eare

st h

un

dre

dth

.

1.�

�15

,m�

36,n

�39

2.�

�12

,m�

12�

3�,n

�24

sin

L�

�1 35 9��

0.38

;si

n L

��1 22 4�

�0.

50;

cos

L�

�3 36 9��

0.92

;co

s L

��

0.87

;

tan

L�

�1 35 6��

0.42

;ta

n L

��

0.58

;

sin

M�

�3 36 9��

0.92

;si

n M

��

0.87

;

cos

M�

�1 35 9��

0.38

;co

s M

��1 22 4�

�0.

50;

tan

M�

�3 16 5��

2.4

tan

M�

�1.

73

Use

a c

alcu

lato

r to

fin

d e

ach

val

ue.

Rou

nd

to

the

nea

rest

ten

-th

ousa

nd

th.

3.si

n 9

2.4

0.99

914.

tan

27.

50.

5206

5.co

s 64

.80.

4258

Use

th

e fi

gure

to

fin

d e

ach

tri

gon

omet

ric

rati

o.E

xpre

ss

answ

ers

as a

fra

ctio

n a

nd

as

a d

ecim

al r

oun

ded

to

the

nea

rest

ten

-th

ousa

nd

th.

6.co

s A

7.ta

n B

8.si

n A

�0.

9487

�3 1��

3.00

00�

0.31

62

Fin

d t

he

mea

sure

of

each

acu

te a

ngl

e to

th

e n

eare

st t

enth

of

a d

egre

e.

9.si

n B

�0.

7823

51.5

10.t

an A

�0.

2356

13.3

11.c

os R

�0.

6401

50.2

Fin

d x

.Rou

nd

to

the

nea

rest

ten

th.

12.

64.4

13.

18.1

14.

24.2

15.G

EOG

RA

PHY

Die

go u

sed

a th

eodo

lite

to

map

a r

egio

n o

f la

nd

for

his

cl

ass

in g

eom

orph

olog

y.T

o de

term

ine

the

elev

atio

n o

f a

vert

ical

roc

kfo

rmat

ion

,he

mea

sure

d th

e di

stan

ce f

rom

th

e ba

se o

f th

e fo

rmat

ion

to

his

pos

itio

n a

nd

the

angl

e be

twee

n t

he

grou

nd

and

the

lin

e of

sig

ht

to

the

top

of t

he

form

atio

n.T

he

dist

ance

was

43

met

ers

and

the

angl

e w

as

36 d

egre

es.W

hat

is

the

hei

ght

of t

he

form

atio

n t

o th

e n

eare

st m

eter

?31

m

36� 43

m

41�

x

3229

x�9

23

x�

11

�10�

�10

3 �10�

�10

15

5 ��10

5 CA

B

12�

3��

12

12�

3��

2412� 12

�3�

12�

3��

24

ML

N

Pra

ctic

e (

Ave

rag

e)

Trig

on

om

etry

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csTr

igo

no

met

ry

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4

©G

lenc

oe/M

cGra

w-H

ill37

3G

lenc

oe G

eom

etry

Lesson 7-4

Pre-

Act

ivit

yH

ow c

an s

urv

eyor

s d

eter

min

e an

gle

mea

sure

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-4

at

the

top

of p

age

364

in y

our

text

book

.

•W

hy

is i

t im

port

ant

to d

eter

min

e th

e re

lati

ve p

osit

ion

s ac

cura

tely

in

nav

igat

ion

? (G

ive

two

poss

ible

rea

son

s.)

Sam

ple

an

swer

s:(1

) To

avo

id c

olli

sio

ns

bet

wee

n s

hip

s,an

d (

2) t

o p

reve

nt

ship

sfr

om

losi

ng

th

eir

bea

rin

gs

and

get

tin

g lo

st a

t se

a.•

Wh

at d

oes

cali

brat

edm

ean

? S

amp

le a

nsw

er:

mar

ked

pre

cise

ly t

op

erm

it a

ccu

rate

mea

sure

men

ts t

o b

e m

ade

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.W

rite

a r

atio

usi

ng

the

side

len

gth

s in

th

e

figu

re t

o re

pres

ent

each

of

the

foll

owin

g tr

igon

omet

ric

rati

os.

A.

sin

N� MM

NP �B

.cos

N� MN

P N�

C.

tan

N�M N

PP �D

.tan

M� MN

P P�

E.

sin

M� MN

P N�F.

cos

M� MM

NP �

2.A

ssu

me

that

you

en

ter

each

of

the

expr

essi

ons

in t

he

list

on

th

e le

ft i

nto

you

r ca

lcu

lato

r.M

atch

eac

h o

f th

ese

expr

essi

ons

wit

h a

des

crip

tion

fro

m t

he

list

on

th

e ri

ght

to t

ell

wh

atyo

u a

re f

indi

ng

wh

en y

ou e

nte

r th

is e

xpre

ssio

n.

P

MN

a.si

n 2

0v

b.

cos

20ii

c.si

n�

10.

8vi

d.

tan

�1

0.8

iiie.

tan

20

ivf.

cos�

10.

8i

i.th

e de

gree

mea

sure

of

an a

cute

an

gle

wh

ose

cosi

ne

is 0

.8

ii.

the

rati

o of

th

e le

ngt

h o

f th

e le

g ad

jace

nt

to t

he

20°

angl

e to

th

ele

ngt

h o

f h

ypot

enu

se i

n a

20°

-70°

-90°

tria

ngl

e

iii.

the

degr

ee m

easu

re o

f an

acu

te a

ngl

e in

a r

igh

t tr

ian

gle

for

wh

ich

th

e ra

tio

of t

he

len

gth

of

the

oppo

site

leg

to

the

len

gth

of

the

adja

cen

t le

g is

0.8

iv.t

he r

atio

of

the

leng

th o

f th

e le

g op

posi

te t

he 2

0°an

gle

to t

hele

ngth

of

the

leg

adja

cent

to

it i

n a

20°-

70°-

90°

tria

ngle

v.th

e ra

tio

of t

he

len

gth

of

the

leg

oppo

site

th

e 20

°an

gle

to t

he

len

gth

of

hyp

oten

use

in

a 2

0°-7

0°-9

0°tr

ian

gle

vi.t

he

degr

ee m

easu

re o

f an

acu

te a

ngl

e in

a r

igh

t tr

ian

gle

for

wh

ich

th

e ra

tio

of t

he

len

gth

of

the

oppo

site

leg

to

the

len

gth

of

the

hyp

oten

use

is

0.8

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an t

he

coin

cos

ine

hel

p yo

u t

o re

mem

ber

the

rela

tion

ship

bet

wee

n t

he

sin

es a

nd

cosi

nes

of

the

two

acu

te a

ngl

es o

f a

righ

t tr

ian

gle?

Sam

ple

an

swer

:Th

e co

in c

osi

ne

com

es f

rom

co

mp

lem

ent,

as in

com

ple

men

tary

ang

les.

Th

e co

sin

e o

f an

acu

te a

ng

le is

eq

ual

to

th

e si

ne

of

its

com

ple

men

t.

©G

lenc

oe/M

cGra

w-H

ill37

4G

lenc

oe G

eom

etry

Sin

e an

d C

osi

ne

of

An

gle

sT

he

foll

owin

g di

agra

m c

an b

e u

sed

to o

btai

n a

ppro

xim

ate

valu

es f

or t

he

sin

ean

d co

sin

e of

an

gles

fro

m 0

°to

90°

.Th

e ra

diu

s of

th

e ci

rcle

is

1.S

o,th

e si

ne

and

cosi

ne

valu

es c

an b

e re

ad d

irec

tly

from

th

e ve

rtic

al a

nd

hor

izon

tal

axes

.

Fin

d a

pp

roxi

mat

e va

lues

for

sin

40°

and

cos

40�

.Con

sid

er t

he

tria

ngl

e fo

rmed

by

the

segm

ent

mar

ked

40°

,as

illu

stra

ted

by

the

shad

ed

tria

ngl

e at

rig

ht.

sin

40°

��a c�

��0.

164 �or

0.6

4co

s 40

°�

�b c��

�0.177 �

or 0

.77

1.U

se t

he

diag

ram

abo

ve t

o co

mpl

ete

the

char

t of

val

ues

.

2.C

ompa

re t

he

sin

e an

d co

sin

e of

tw

o co

mpl

emen

tary

an

gles

(an

gles

wh

ose

sum

is

90°)

.Wh

at d

o yo

u n

otic

e?

Th

e si

ne

of

an a

ng

le is

eq

ual

to

th

e co

sin

e o

f th

e co

mp

lem

ent

of

the

ang

le.

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

90°

0°

10°

20°

30°

40°

50°

60°

70°

80°

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-4

7-4 x

°0°

10°

20°

30°

40°

50°

60°

70°

80°

90°

sin

x°

00.

170.

340.

50.

640.

770.

870.

940.

981

cos

x°

10.

980.

940.

870.

770.

640.

50.

340.

170

1 0

40°

0.64

c �

1 u

nit

x°b

� c

os x

°0.

771

a �

sin

x°

Exam

ple

Exam

ple



Stu

dy G

uid

e a

nd I

nte

rven

tion

An

gle

s o

f E

leva

tio

n a

nd

Dep

ress

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-5

7-5

©G

lenc

oe/M

cGra

w-H

ill37

5G

lenc

oe G

eom

etry

Lesson 7-5

An

gle

s o

f El

evat

ion

Man

y re

al-w

orld

pro

blem

s th

at i

nvo

lve

look

ing

up

to a

n o

bjec

t ca

n b

e de

scri

bed

in t

erm

s of

an

an

gle

of

elev

atio

n,w

hic

h i

s th

e an

gle

betw

een

an

obs

erve

r’s

lin

e of

sig

ht

and

a h

oriz

onta

l li

ne.

Th

e an

gle

of e

leva

tion

fro

m p

oin

t A

to t

he

top

of

a c

liff

is

34°.

If p

oin

t A

is 1

000

feet

fro

m t

he

bas

e of

th

e cl

iff,

how

hig

h i

s th

e cl

iff?

Let

x�

the

hei

ght

of t

he

clif

f.

tan

34°

�� 10

x 00�ta

n �

�o ap dp jao cs eit ne t�

1000

(tan

34°

)�

xM

ultip

ly e

ach

side

by

1000

.

674.

5�

xU

se a

cal

cula

tor.

Th

e h

eigh

t of

th

e cl

iff

is a

bou

t 67

4.5

feet

.

Sol

ve e

ach

pro

ble

m.R

oun

d m

easu

res

of s

egm

ents

to

the

nea

rest

wh

ole

nu

mb

eran

d a

ngl

es t

o th

e n

eare

st d

egre

e.

1.T

he

angl

e of

ele

vati

on f

rom

poi

nt

Ato

th

e to

p of

a h

ill

is 4

9°.

If p

oin

t A

is 4

00 f

eet

from

th

e ba

se o

f th

e h

ill,

how

hig

h i

s th

e h

ill?

460

ft

2.F

ind

the

angl

e of

ele

vati

on o

f th

e su

n w

hen

a 1

2.5-

met

er-t

all

tele

phon

e po

le c

asts

a 1

8-m

eter

-lon

g sh

adow

.

35°

3.A

lad

der

lean

ing

agai

nst

a b

uil

din

g m

akes

an

an

gle

of 7

8°w

ith

th

e gr

oun

d.T

he

foot

of

the

ladd

er i

s 5

feet

fro

m t

he

buil

din

g.H

ow l

ong

is t

he

ladd

er?

24 f

t

4.A

per

son

wh

ose

eyes

are

5 f

eet

abov

e th

e gr

oun

d is

sta

ndi

ng

on t

he

run

way

of

an a

irpo

rt 1

00 f

eet

from

th

e co

ntr

ol t

ower

.T

hat

per

son

obs

erve

s an

air

tra

ffic

con

trol

ler

at t

he

win

dow

of

th

e 13

2-fo

ot t

ower

.Wh

at i

s th

e an

gle

of e

leva

tion

?

52°

?5

ft10

0 ft

132

ft

78�

5 ft

?

18 m

12.5

msun

?

✹

400

ft

?

49�

A

?

1000

ft34

�A

x

angl

e of

elev

atio

n

line o

f sigh

t

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill37

6G

lenc

oe G

eom

etry

An

gle

s o

f D

epre

ssio

nW

hen

an

obs

erve

r is

loo

kin

g do

wn

,th

e an

gle

of d

epre

ssio

nis

th

e an

gle

betw

een

th

e ob

serv

er’s

lin

e of

sig

ht

and

a h

oriz

onta

l li

ne.

Th

e an

gle

of d

epre

ssio

n f

rom

th

e to

p o

f an

80

-foo

t b

uil

din

g to

poi

nt

Aon

th

e gr

oun

d i

s 42

°.H

ow f

ar

is t

he

foot

of

the

bu

ild

ing

from

poi

nt

A?

Let

x�

the

dist

ance

fro

m p

oin

t A

to t

he

foot

of

the

buil

din

g.S

ince

th

e h

oriz

onta

l li

ne

is p

aral

lel

to t

he

grou

nd,

the

angl

e of

dep

ress

ion

�D

BA

is c

ongr

uen

t to

�B

AC

.

tan

42°

��8 x0 �

tan

��o ap dp jao cs eit ne t

�

x(ta

n 4

2°)

�80

Mul

tiply

eac

h si

de b

y x.

x�

� tan80

42°

�D

ivid

e ea

ch s

ide

by t

an 4

2°.

x�

88.8

Use

a c

alcu

lato

r.

Poi

nt

Ais

abo

ut

89 f

eet

from

th

e ba

se o

f th

e bu

ildi

ng.

Sol

ve e

ach

pro

ble

m.R

oun

d m

easu

res

of s

egm

ents

to

the

nea

rest

wh

ole

nu

mb

eran

d a

ngl

es t

o th

e n

eare

st d

egre

e.

1.T

he

angl

e of

dep

ress

ion

fro

m t

he

top

of a

sh

eer

clif

f to

po

int

Aon

th

e gr

oun

d is

35°

.If

poin

t A

is 2

80 f

eet

from

th

e ba

se o

f th

e cl

iff,

how

tal

l is

th

e cl

iff?

196

ft

2.T

he

angl

e of

dep

ress

ion

fro

m a

bal

loon

on

a 7

5-fo

ot

stri

ng

to a

per

son

on

th

e gr

oun

d is

36°

.How

hig

h i

s th

e ba

lloo

n?

44 f

t

3.A

ski

ru

n i

s 10

00 y

ards

lon

g w

ith

a v

erti

cal

drop

of

208

yard

s.F

ind

the

angl

e of

dep

ress

ion

fro

m t

he

top

of t

he

ski

run

to

the

bott

om.

12°

4.F

rom

th

e to

p of

a 1

20-f

oot-

hig

h t

ower

,an

air

tra

ffic

co

ntr

olle

r ob

serv

es a

n a

irpl

ane

on t

he

run

way

at

an

angl

e of

dep

ress

ion

of

19°.

How

far

fro

m t

he

base

of

the

tow

er i

s th

e ai

rpla

ne?

349

ft

120

ft

?

19�

208

yd

?

1000

yd

36�

75 ft

?

A

35�

280

ft

?

ACB

D

x42

�

angl

e of

depr

essi

on

horiz

onta

l

80 ft

Ylin

e of s

ight

horiz

onta

lan

gle

ofde

pres

sion

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

An

gle

s o

f E

leva

tio

n a

nd

Dep

ress

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-5

7-5

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

An

gle

s o

f E

leva

tio

n a

nd

Dep

ress

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-5

7-5

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eom

etry

Lesson 7-5

Nam

e th

e an

gle

of d

epre

ssio

n o

r an

gle

of e

leva

tion

in

eac

h f

igu

re.

1.2.

�F

LS

;�

TS

L�

RT

W;

�S

WT

3.4.

�D

CB

;�

AB

C�

WZ

P;

�R

PZ

5.M

OU

NTA

IN B

IKIN

GO

n a

mou

nta

in b

ike

trip

alo

ng

the

Gem

ini

Bri

dges

Tra

il i

n M

oab,

Uta

h,N

abu

ko s

topp

ed o

n t

he

can

yon

flo

or t

o ge

t a

good

vie

w o

f th

e tw

in s

ands

ton

ebr

idge

s.N

abu

ko i

s st

andi

ng

abou

t 60

met

ers

from

th

e ba

se o

f th

e ca

nyo

n c

liff

,an

d th

en

atu

ral

arch

bri

dges

are

abo

ut

100

met

ers

up

the

can

yon

wal

l.If

her

lin

e of

sig

ht

is f

ive

feet

abo

ve t

he

grou

nd,

wh

at i

s th

e an

gle

of e

leva

tion

to

the

top

of t

he

brid

ges?

Rou

nd

toth

e n

eare

st t

enth

deg

ree.

abo

ut

57.7

�

6.SH

AD

OW

SS

upp

ose

the

sun

cas

ts a

sh

adow

off

a 3

5-fo

ot b

uil

din

g.If

th

e an

gle

of e

leva

tion

to

the

sun

is

60°,

how

lon

g is

th

e sh

adow

to

th

e n

eare

st t

enth

of

a fo

ot?

abo

ut

20.2

ft

7.B

ALL

OO

NIN

GF

rom

her

pos

itio

n i

n a

hot

-air

bal

loon

,An

gie

can

see

her

car

par

ked

in a

fiel

d.If

th

e an

gle

of d

epre

ssio

n i

s 8°

and

An

gie

is 3

8 m

eter

s ab

ove

the

grou

nd,

wh

at i

sth

e st

raig

ht-

lin

e di

stan

ce f

rom

An

gie

to h

er c

ar?

Rou

nd

to t

he

nea

rest

wh

ole

met

er.

abo

ut

273

m

8.IN

DIR

ECT

MEA

SUR

EMEN

TK

yle

is a

t th

e en

d of

a p

ier

30 f

eet

abov

e th

e oc

ean

.His

eye

lev

el i

s 3

feet

abo

ve t

he

pier

.He

is u

sin

g bi

noc

ula

rs t

o w

atch

a w

hal

e su

rfac

e.If

th

e an

gle

of d

epre

ssio

n

of t

he

wh

ale

is 2

0°,h

ow f

ar i

s th

e w

hal

e fr

om

Kyl

e’s

bin

ocu

lars

? R

oun

d to

th

e n

eare

st t

enth

foo

t.

abo

ut

96.5

ft

wha

lew

ater

leve

l

20�

Kyle

’s ey

es

pier

3 ft

30 ft

60� ?

35 ft

Z

PW

R

D

AC

B

T

WR

S

F

T

L

S

©G

lenc

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ill37

8G

lenc

oe G

eom

etry

Nam

e th

e an

gle

of d

epre

ssio

n o

r an

gle

of e

leva

tion

in

eac

h f

igu

re.

1.2.

�T

RZ

;�

YZ

R�

PR

M;

�L

MR

3.W

ATE

R T

OW

ERS

A s

tude

nt

can

see

a w

ater

tow

er f

rom

th

e cl

oses

t po

int

of t

he

socc

erfi

eld

at S

an L

obos

Hig

h S

choo

l.T

he

edge

of

the

socc

er f

ield

is

abou

t 11

0 fe

et f

rom

th

ew

ater

tow

er a

nd

the

wat

er t

ower

sta

nds

at

a h

eigh

t of

32.

5 fe

et.W

hat

is

the

angl

e of

elev

atio

n i

f th

e ey

e le

vel

of t

he

stu

den

t vi

ewin

g th

e to

wer

fro

m t

he

edge

of

the

socc

erfi

eld

is 6

fee

t ab

ove

the

grou

nd?

Rou

nd

to t

he

nea

rest

ten

th d

egre

e.

abo

ut

13.5

�

4.C

ON

STR

UC

TIO

NA

roo

fer

prop

s a

ladd

er a

gain

st a

wal

l so

th

at t

he

top

of t

he

ladd

erre

ach

es a

30-

foot

roo

f th

at n

eeds

rep

air.

If t

he

angl

e of

ele

vati

on f

rom

th

e bo

ttom

of

the

ladd

er t

o th

e ro

of i

s 55

°,h

ow f

ar i

s th

e la

dder

fro

m t

he

base

of

the

wal

l? R

oun

d yo

ur

answ

er t

o th

e n

eare

st f

oot.

abo

ut

21 f

t

5.TO

WN

OR

DIN

AN

CES

Th

e to

wn

of

Bel

mon

t re

stri

cts

the

hei

ght

of f

lagp

oles

to

25 f

eet

on a

ny

prop

erty

.Lin

dsay

wan

ts t

o de

term

ine

wh

eth

er h

er s

choo

l is

in

com

plia

nce

wit

h t

he

regu

lati

on.H

er e

ye

leve

l is

5.5

fee

t fr

om t

he

grou

nd

and

she

stan

ds 3

6 fe

et f

rom

th

efl

agpo

le.I

f th

e an

gle

of e

leva

tion

is

abou

t 25

°,w

hat

is

the

hei

ght

of t

he

flag

pole

to

the

nea

rest

ten

th f

oot?

abo

ut

22.3

ft

6.G

EOG

RA

PHY

Ste

phan

is

stan

din

g on

a m

esa

at t

he

Pai

nte

d D

eser

t.T

he

elev

atio

n o

fth

e m

esa

is a

bou

t 13

80 m

eter

s an

d S

teph

an’s

eye

lev

el i

s 1.

8 m

eter

s ab

ove

grou

nd.

IfS

teph

an c

an s

ee a

ban

d of

mu

ltic

olor

ed s

hal

e at

th

e bo

ttom

an

d th

e an

gle

of d

epre

ssio

nis

29°

,abo

ut

how

far

is

the

ban

d of

sh

ale

from

his

eye

s? R

oun

d to

th

e n

eare

st m

eter

.

abo

ut

2850

m

7.IN

DIR

ECT

MEA

SUR

EMEN

TM

r.D

omin

guez

is

stan

din

g on

a 4

0-fo

ot o

cean

blu

ff n

ear

his

hom

e.H

e ca

n s

ee h

is t

wo

dogs

on

th

e be

ach

bel

ow.I

f h

is l

ine

of s

igh

t is

6 f

eet

abov

e th

e gr

oun

d an

d th

e an

gles

of

depr

essi

on t

o h

is d

ogs

are

34°

and

48°,

how

far

apa

rt a

re t

he

dogs

to

the

nea

rest

foo

t?

abo

ut

27 f

t48

�34

�

40 ft

6 ft

Mr.

Dom

ingu

ez

bluf

f

25�

5.5

ft36

ft

x

R

M

P

L

T

YR

Z

Pra

ctic

e (

Ave

rag

e)

An

gle

s o

f E

leva

tio

n a

nd

Dep

ress

ion

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-5

7-5



Readin

g t

o L

earn

Math

em

ati

csA

ng

les

of

Ele

vati

on

an

d D

epre

ssio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-5

7-5

©G

lenc

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ill37

9G

lenc

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eom

etry

Lesson 7-5

Pre-

Act

ivit

yH

ow d

o ai

rlin

e p

ilot

s u

se a

ngl

es o

f el

evat

ion

an

d d

epre

ssio

n?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-5

at

the

top

of p

age

371

in y

our

text

book

.

Wh

at d

oes

the

angl

e m

easu

re t

ell

the

pilo

t?S

amp

le a

nsw

er:

ho

wst

eep

her

asc

ent

mu

st b

e to

cle

ar t

he

pea

k

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.T

he

two

obse

rver

s ar

e lo

okin

g at

on

e an

othe

r.S

elec

t th

e co

rrec

t ch

oice

for

eac

h qu

esti

on.

a.W

hat

is

the

lin

e of

sig

ht?

iii(i

) li

ne

RS

(ii)

lin

e S

T(i

ii)

lin

e R

T(i

v) l

ine

TU

b.

Wh

at i

s th

e an

gle

of e

leva

tion

?ii

(i)

�R

ST

(ii)

�S

RT

(iii

) �

RT

S(i

v) �

UT

R

c.W

hat

is

the

angl

e of

dep

ress

ion

?iv

(i)

�R

ST

(ii)

�S

RT

(iii

) �

RT

S(i

v) �

UT

R

d.

How

are

th

e an

gle

of e

leva

tion

an

d th

e an

gle

of d

epre

ssio

n r

elat

ed?

ii(i

)T

hey

are

com

plem

enta

ry.

(ii)

Th

ey a

re c

ongr

uen

t.(i

ii)

Th

ey a

re s

upp

lem

enta

ry.

(iv)

Th

e an

gle

of e

leva

tion

is

larg

er t

han

th

e an

gle

of d

epre

ssio

n.

e.W

hic

h p

ostu

late

or

theo

rem

th

at y

ou l

earn

ed i

n C

hap

ter

3 su

ppor

ts y

our

answ

er f

orpa

rt c

?iv

(i)

Cor

resp

ondi

ng

An

gles

Pos

tula

te(i

i)A

lter

nat

e E

xter

ior

An

gles

Th

eore

m(i

ii)

Con

secu

tive

In

teri

or A

ngl

es T

heo

rem

(iv)

Alt

ern

ate

Inte

rior

An

gles

Th

eore

m

2.A

stu

den

t sa

ys t

hat

th

e an

gle

of e

leva

tion

fro

m h

is e

ye t

o th

e to

p of

a f

lagp

ole

is 1

35°.

Wh

at i

s w

ron

g w

ith

th

e st

ude

nt’s

sta

tem

ent?

An

an

gle

of

elev

atio

n c

ann

ot

be

ob

tuse

.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

som

eth

ing

is t

o ex

plai

n i

t to

som

eon

e el

se.S

upp

ose

a cl

assm

ate

find

s it

dif

ficu

lt t

o di

stin

guis

h be

twee

n an

gles

of

elev

atio

n an

d an

gles

of

depr

essi

on.W

hat

are

som

e h

ints

you

can

giv

e h

er t

o h

elp

her

get

it

righ

t ev

ery

tim

e?S

amp

le a

nsw

ers:

(1) T

he

ang

le o

f d

epre

ssio

n a

nd

th

e an

gle

of

elev

atio

n a

re b

oth

mea

sure

db

etw

een

th

e h

ori

zon

tal a

nd

th

e lin

e o

f si

gh

t.(2

) Th

e an

gle

of

dep

ress

ion

is a

lway

s co

ng

ruen

t to

th

e an

gle

of

elev

atio

n in

th

e sa

me

dia

gra

m.

(3)

Ass

oci

ate

the

wo

rd e

leva

tio

nw

ith

th

e w

ord

up

and

th

e w

ord

dep

ress

ion

wit

h t

he

wo

rd d

own

.

STob

serv

er a

tto

p of

bui

ldin

g

obse

rver

on g

roun

dR

U

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Rea

din

g M

ath

emat

ics

Th

e th

ree

mos

t co

mm

on t

rigo

nom

etri

c ra

tios

are

si

ne,

cosi

ne,

and

tan

gen

t.T

hre

e ot

her

rat

ios

are

the

cose

can

t,se

can

t,an

d co

tan

gen

t.T

he

char

t be

low

sh

ows

abbr

evia

tion

s an

d de

fin

itio

ns

for

all

six

rati

os.

Ref

er t

o th

e tr

ian

gle

at t

he

righ

t.

Use

th

e ab

bre

viat

ion

s to

rew

rite

eac

h s

tate

men

t as

an

eq

uat

ion

.

1.T

he

seca

nt

of a

ngl

e A

is

equ

al t

o 1

divi

ded

by t

he

cosi

ne

of a

ngl

e A

.se

c A

�� co

1 sA

�

2.T

he

cose

can

t of

an

gle

A i

s eq

ual

to

1 di

vide

d by

th

e si

ne

of a

ngl

e A

.cs

c A

�� si

n1A�

3.T

he

cota

nge

nt

of a

ngl

e A

is

equ

al t

o 1

divi

ded

by t

he

tan

gen

t of

an

gle

A.

cot

A�

� tan1

A�

4.T

he c

osec

ant

of a

ngle

A m

ulti

plie

d by

the

sin

e of

ang

le A

is e

qual

to

1.cs

c A

sin

A�

1

5.T

he s

ecan

t of

ang

le A

mul

tipl

ied

by t

he c

osin

e of

ang

le A

is e

qual

to

1.se

c A

co

s A

�1

6.T

he

cota

nge

nt

of a

ngl

e A

tim

es t

he

tan

gen

t of

an

gle

Ais

equ

al t

o 1.

cot

A t

an A

�1

Use

th

e tr

ian

gle

at r

igh

t.W

rite

eac

h r

atio

.

7.se

c R

� st �8.

csc

R� rt �

9.co

t R

�s r�

10.s

ec S

� rt �11

.cs

c S

� st �12

.co

t S

� sr �

13.I

f si

n x

°�

0.28

9,fi

nd

the

valu

e of

csc

x°.

�3.

46

14.I

f ta

n x

°�

1.37

6,fi

nd

the

valu

e of

cot

x°.

�0.

727

R TS

ts

r

A

ca

b

B C

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-5

7-5

Ab

bre

viat

ion

Rea

d a

s:R

atio

sin

Ath

e si

ne o

f �

A�

�a c�

cos

Ath

e co

sine

of

�A

��b c�

tan

Ath

e ta

ngen

t of

�A

��a b�

csc

Ath

e co

seca

nt o

f �

A�

� ac �

sec

Ath

e se

cant

of

�A

�� bc �

cot

Ath

e co

tang

ent

of �

A�

�b a�le

gad

jace

nt t

o�

A�

��

leg

oppo

site

�A

hypo

tenu

se�

��

leg

adja

cent

to

�A

hypo

tenu

se�

�le

gop

posi

te�

A

leg

oppo

site

�A

��

�le

gad

jace

nt t

o�

A

leg

adja

cent

to

�A

��

�hy

pote

nuse

leg

oppo

site

�A

��

hypo

tenu

se



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Th

e L

aw o

f S

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

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ER

IOD

____

_

7-6

7-6

©G

lenc

oe/M

cGra

w-H

ill38

1G

lenc

oe G

eom

etry

Lesson 7-6

The

Law

of

Sin

esIn

an

y tr

ian

gle,

ther

e is

a s

peci

al r

elat

ion

ship

bet

wee

n t

he

angl

es o

fth

e tr

ian

gle

and

the

len

gth

s of

th

e si

des

oppo

site

th

e an

gles

.

Law

of

Sin

es�si

n aA �

��si

n bB �

��si

n cC �

In �

AB

C,f

ind

b.

�sin c

C ��

�sin b

B �La

w o

f S

ines

�sin 30

45°

��

�sin b74

°�

m�

C�

45, c

�30

, m�

B�

74

bsi

n 4

5°�

30 s

in 7

4°C

ross

mul

tiply.

b�

�30si

s nin 457 °4°

�D

ivid

e ea

ch s

ide

by s

in 4

5°.

b�

40.8

Use

a c

alcu

lato

r.

45�

3074

�

bB

AC

In �

DE

F,f

ind

m�

D.

�sin d

D ��

�sin e

E �La

w o

f S

ines

�si2n 8D �

��si

n 2458

°�

d�

28

, m

�E

�5

8,

e�

24

24 s

in D

�28

sin

58°

Cro

ss m

ultip

ly.

sin

D�

�28s 2in 4

58°

�D

ivid

e ea

ch s

ide

by 2

4.

D�

sin

�1 �28

s 2in 458

°�

Use

the

inve

rse

sine

.

D�

81.6

°U

se a

cal

cula

tor.

58�

24

28

E

FD

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

of �

AB

C.R

oun

d a

ngl

e m

easu

res

toth

e n

eare

st d

egre

e an

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

1.If

c�

12,m

�A

�80

,an

d m

�C

�40

,fin

d a.

18.4

2.If

b�

20,c

�26

,an

d m

�C

�52

,fin

d m

�B

.

37

3.If

a�

18,c

�16

,an

d m

�A

�84

,fin

d m

�C

.

62

4.If

a�

25,m

�A

�72

,an

d m

�B

�17

,fin

d b.

7.7

5.If

b�

12,m

�A

�89

,an

d m

�B

�80

,fin

d a.

12.2

6.If

a�

30,c

�20

,an

d m

�A

�60

,fin

d m

�C

.

35

©G

lenc

oe/M

cGra

w-H

ill38

2G

lenc

oe G

eom

etry

Use

th

e La

w o

f Si

nes

to

So

lve

Pro

ble

ms

You

can

use

th

e L

aw o

f S

ines

to s

olve

som

e pr

oble

ms

that

in

volv

e tr

ian

gles

.

Law

of

Sin

esLe

t �

AB

Cbe

any

tria

ngle

with

a,

b, a

nd c

repr

esen

ting

the

mea

sure

s of

the

sid

es o

ppos

ite

the

angl

es w

ith m

easu

res

A,

B,

and

C,

resp

ectiv

ely.

The

n �si

n aA �

��si

n bB �

��si

n cC �

.

Isos

cele

s �

AB

Ch

as a

bas

e of

24

cen

tim

eter

s an

d a

ve

rtex

an

gle

of 6

8°.F

ind

th

e p

erim

eter

of

the

tria

ngl

e.T

he

vert

ex a

ngl

e is

68°

,so

the

sum

of

the

mea

sure

s of

th

e ba

se a

ngl

es i

s 11

2 an

d m

�A

�m

�C

�56

.

�sin b

B ��

�sin a

A�

Law

of

Sin

es

�sin 24

68°

��

�sin a56

°�

m�

B�

68,

b�

24,

m�

A�

56

asi

n 6

8°�

24 s

in 5

6°C

ross

mul

tiply

.

a�

�24si

s nin 685 °6°

�D

ivid

e ea

ch s

ide

by s

in 6

8°.

�21

.5U

se a

cal

cula

tor.

Th

e tr

ian

gle

is i

sosc

eles

,so

c�

21.5

.T

he

peri

met

er i

s 24

�21

.5 �

21.5

or

abou

t 67

cen

tim

eter

s.

Dra

w a

tri

angl

e to

go

wit

h e

ach

exe

rcis

e an

d m

ark

it

wit

h t

he

give

n i

nfo

rmat

ion

.T

hen

sol

ve t

he

pro

ble

m.R

oun

d a

ngl

e m

easu

res

to t

he

nea

rest

deg

ree

and

sid

em

easu

res

to t

he

nea

rest

ten

th.

1.O

ne

side

of

a tr

ian

gula

r ga

rden

is

42.0

fee

t.T

he

angl

es o

n e

ach

en

d of

th

is s

ide

mea

sure

66°

and

82°.

Fin

d th

e le

ngt

h o

f fe

nce

nee

ded

to e

ncl

ose

the

gard

en.

192.

9 ft

2.T

wo

rada

r st

atio

ns

Aan

d B

are

32 m

iles

apa

rt.T

hey

loc

ate

an a

irpl

ane

Xat

th

e sa

me

tim

e.T

he

thre

e po

ints

for

m �

XA

B,w

hic

h m

easu

res

46°,

and

�X

BA

,wh

ich

mea

sure

s52

°.H

ow f

ar i

s th

e ai

rpla

ne

from

eac

h s

tati

on?

25.5

mi f

rom

A;

23.2

mi f

rom

B

3.A

civ

il e

ngi

nee

r w

ants

to

dete

rmin

e th

e di

stan

ces

from

poi

nts

Aan

d B

to a

n i

nac

cess

ible

poin

t C

in a

riv

er.�

BA

Cm

easu

res

67°

and

�A

BC

mea

sure

s 52

°.If

poi

nts

Aan

d B

are

82.0

fee

t ap

art,

fin

d th

e di

stan

ce f

rom

Cto

eac

h p

oin

t.

86.3

ft

to p

oin

t B

;73

.9 f

t to

po

int

A

4.A

ran

ger

tow

er a

t po

int

Ais

42

kilo

met

ers

nor

th o

f a

ran

ger

tow

er a

t po

int

B.A

fir

e at

poin

t C

is o

bser

ved

from

bot

h t

ower

s.If

�B

AC

mea

sure

s 43

°an

d �

AB

Cm

easu

res

68°,

wh

ich

ran

ger

tow

er i

s cl

oser

to

the

fire

? H

ow m

uch

clo

ser?

Tow

er B

is 1

1 km

clo

ser

than

To

wer

A.

68� b

ca

24B

CA

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Th

e L

aw o

f S

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-6

7-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Th

e L

aw o

f S

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-6

7-6

©G

lenc

oe/M

cGra

w-H

ill38

3G

lenc

oe G

eom

etry

Lesson 7-6

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

from

�A

BC

.Rou

nd

an

gle

mea

sure

sto

th

e n

eare

st t

enth

deg

ree

and

sid

e m

easu

res

to t

he

nea

rest

ten

th.

1.If

m�

A�

35,m

�B

�48

,an

d b

�28

,fin

d a.

21.6

2.If

m�

B�

17,m

�C

�46

,an

d c

�18

,fin

d b.

7.3

3.If

m�

C�

86,m

�A

�51

,an

d a

�38

,fin

d c.

48.8

4.If

a�

17,b

�8,

and

m�

A�

73,f

ind

m�

B.

26.7

5.If

c�

38,b

�34

,an

d m

�B

�36

,fin

d m

�C

.41

.1 o

r 13

8.9

6.If

a�

12,c

�20

,an

d m

�C

�83

,fin

d m

�A

.36

.6

7.If

m�

A�

22,a

�18

,an

d m

�B

�10

4,fi

nd

b.46

.6

Sol

ve e

ach

�P

QR

des

crib

ed b

elow

.Rou

nd

mea

sure

s to

th

e n

eare

st t

enth

.

8.p

�27

,q�

40,m

�P

�33

m�

Q�

53.8

,m�

R�

93.2

,r�

49.5

;o

r m

�Q

�12

6.2,

m�

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20.8

,r�

17.6

9.q

�12

,r�

11,m

�R

�16

m�

P�

146.

5,m

�Q

�17

.5,p

�22

.0;

or

m�

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1.5,

m�

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162.

5,p

�1.

0

10.p

�29

,q�

34,m

�Q

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f m

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10.8

12.I

f m

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3,m

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13m

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14.2

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3.5

13.I

f m

�P

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R�

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m�

Q�

2,p

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�1.

2

14.I

f m

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Q�

76,r

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m�

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f m

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1

©G

lenc

oe/M

cGra

w-H

ill38

4G

lenc

oe G

eom

etry

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

from

�E

FG

.Rou

nd

an

gle

mea

sure

sto

th

e n

eare

st t

enth

deg

ree

and

sid

e m

easu

res

to t

he

nea

rest

ten

th.

1.If

m�

G�

14,m

�E

�67

,an

d e

�14

,fin

d g.

3.7

2.If

e�

12.7

,m�

E�

42,a

nd

m�

F�

61,f

ind

f.16

.6

3.If

g�

14,f

�5.

8,an

d m

�G

�83

,fin

d m

�F

.24

.3

4.If

e�

19.1

,m�

G�

34,a

nd

m�

E�

56,f

ind

g.12

.9

5.If

f�

9.6,

g�

27.4

,an

d m

�G

�43

,fin

d m

�F

.13

.8

Sol

ve e

ach

�S

TU

des

crib

ed b

elow

.Rou

nd

mea

sure

s to

th

e n

eare

st t

enth

.

6.m

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�85

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4.3,

t�

8.2

m�

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31.5

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63.5

,u�

7.4

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m�

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132.

6,m

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8.m

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2.3,

m�

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17m

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6,s

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4,u

�4.

7

9.m

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�62

,m�

U�

59,s

�17

.8m

�T

�59

,t�

17.3

,u�

17.3

10.t

�28

.4,u

�21

.7,m

�T

�66

m�

S�

69.7

,m�

U�

44.3

,s�

29.2

11.m

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�89

,s�

15.3

,t�

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4

12.m

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U�

74,u

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6m

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s�

1.4,

t�

9.9

13.t

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�S

�84

,m�

T�

47m

�U

�49

,s�

16.0

,u�

12.2

14.I

ND

IREC

T M

EASU

REM

ENT

To

fin

d th

e di

stan

ce f

rom

th

e ed

ge

of t

he

lake

to

the

tree

on

th

e is

lan

d in

th

e la

ke,H

ann

ah s

et u

p a

tria

ngu

lar

con

figu

rati

on a

s sh

own

in

th

e di

agra

m.T

he

dist

ance

fr

om l

ocat

ion

Ato

loc

atio

n B

is 8

5 m

eter

s.T

he

mea

sure

s of

th

e an

gles

at

Aan

d B

are

51°

and

83°,

resp

ecti

vely

.Wha

t is

the

dis

tanc

efr

om t

he

edge

of

the

lake

at

Bto

th

e tr

ee o

n t

he

isla

nd

at C

?

abo

ut

91.8

m

A

C

B

Pra

ctic

e (

Ave

rag

e)

Th

e L

aw o

f S

ines

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-6

7-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csT

he

Law

of

Sin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-6

7-6

©G

lenc

oe/M

cGra

w-H

ill38

5G

lenc

oe G

eom

etry

Lesson 7-6

Pre-

Act

ivit

yH

ow a

re t

rian

gles

use

d i

n r

adio

ast

ron

omy?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-6

at

the

top

of p

age

377

in y

our

text

book

.

Wh

y m

igh

t se

vera

l an

ten

nas

be

bett

er t

han

on

e si

ngl

e an

ten

na

wh

enst

udy

ing

dist

ant

obje

cts?

Sam

ple

an

swer

:O

bse

rvin

g a

n o

bje

ctfr

om

on

ly o

ne

po

siti

on

oft

en d

oes

no

t p

rovi

de

eno

ug

hin

form

atio

n t

o c

alcu

late

th

ing

s su

ch a

s th

e d

ista

nce

fro

m t

he

ob

serv

er t

o t

he

ob

ject

.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.A

ccor

din

g to

th

e L

aw o

f S

ines

,wh

ich

of

the

fo

llow

ing

are

corr

ect

stat

emen

ts?

A,F

A.� si

nmM�

�� si

nnN�

�� si

npP

�B

.�si

n Mm �

��si

Nnn

��

�sin P

p�

C.�co

msM �

��co

s nN �

��co

psP

�D

.�si

n mM �

��si

n nN �

��si

n pP

�

E.

(sin

M)2

�(s

in N

)2�

(sin

P)2

F.�si

n pP

��

�sin m

M ��

�sin n

N �

2.S

tate

wh

eth

er e

ach

of

the

foll

owin

g st

atem

ents

is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,ex

plai

n w

hy.

a.T

he

Law

of

Sin

es a

ppli

es t

o al

l tr

ian

gles

.tr

ue

b.

Th

e P

yth

agor

ean

Th

eore

m a

ppli

es t

o al

l tr

ian

gles

.Fa

lse;

sam

ple

an

swer

:It

on

ly a

pp

lies

to r

igh

t tr

ian

gle

s.c.

If y

ou a

re g

iven

th

e le

ngt

h o

f on

e si

de o

f a

tria

ngl

e an

d th

e m

easu

res

of a

ny

two

angl

es,y

ou c

an u

se t

he

Law

of

Sin

es t

o fi

nd

the

len

gth

s of

th

e ot

her

tw

o si

des.

tru

ed

.If

you

kn

ow t

he

mea

sure

s of

tw

o an

gles

of

a tr

ian

gle,

you

sh

ould

use

th

e L

aw o

f S

ines

to f

ind

the

mea

sure

of

the

thir

d an

gle.

Fals

e;sa

mp

le a

nsw

er:Y

ou

sh

ou

ld u

seth

e A

ng

le S

um

Th

eore

m.

e.A

fri

end

tell

s yo

u t

hat

in

tri

angl

e R

ST

,m�

R�

132,

r�

24 c

enti

met

ers,

and

s�

31ce

nti

met

ers.

Can

you

use

th

e L

aw o

f S

ines

to

solv

e th

e tr

ian

gle?

Exp

lain

.N

o;

sam

ple

an

swer

:In

any

tri

ang

le,t

he

lon

ges

t si

de

is o

pp

osi

te t

he

larg

est

ang

le.B

ecau

se a

tri

ang

le c

an h

ave

on

ly o

ne

ob

tuse

an

gle

,�R

mu

st b

eth

e la

rges

t an

gle

,bu

t s

�r,

so it

is im

po

ssib

le t

o h

ave

a tr

ian

gle

wit

hth

e g

iven

mea

sure

s.

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts r

emem

ber

mat

hem

atic

al e

quat

ion

s an

d fo

rmu

las

bett

er i

f th

ey c

an s

tate

them

in

wor

ds.S

tate

th

e L

aw o

f S

ines

in

you

r ow

n w

ords

wit

hou

t u

sin

g va

riab

les

orm

ath

emat

ical

sym

bols

.S

amp

le a

nsw

er:

In a

ny t

rian

gle

,th

e ra

tio

of

the

sin

e o

f an

an

gle

to

th

ele

ng

th o

f th

e o

pp

osi

te s

ide

is t

he

sam

e fo

r al

l th

ree

ang

les.

P

MN

p

mn

©G

lenc

oe/M

cGra

w-H

ill38

6G

lenc

oe G

eom

etry

Iden

titi

esA

n i

den

tity

is a

n e

quat

ion

th

at i

s tr

ue

for

all

valu

es o

f th

e va

riab

le f

or w

hic

h b

oth

sid

es a

re d

efin

ed.O

ne

way

to

veri

fy

an i

den

tity

is

to u

se a

rig

ht

tria

ngl

e an

d th

e de

fin

itio

ns

for

trig

onom

etri

c fu

nct

ion

s.

Ver

ify

that

(si

n A

)2�

(cos

A)2

�1

is a

n i

den

tity

.

(sin

A)2

�(c

os A

)2�

��a c� �2�

��b c� �2

��a2

� cb2

��

�c c2 2��

1

To

chec

k w

het

her

an

equ

atio

n m

aybe

an

ide

nti

ty,y

ou c

an t

est

seve

ral

valu

es.H

owev

er,s

ince

you

can

not

tes

t al

l va

lues

,you

can

not

be

cert

ain

that

th

e eq

uat

ion

is

an i

den

tity

.

Tes

t si

n 2

x�

2 si

n x

cos

xto

see

if

it c

ould

be

an i

den

tity

.

Try

x�

20.U

se a

cal

cula

tor

to e

valu

ate

each

exp

ress

ion

.

sin

2x

�si

n 4

02

sin

xco

s x

�2

(sin

20)

(cos

20)

�0.

643

�2(

0.34

2)(0

.940

)�

0.64

3

Sin

ce t

he

left

an

d ri

ght

side

s se

em e

qual

,th

e eq

uat

ion

may

be

an i

den

tity

.

Use

tri

angl

e A

BC

show

n a

bov

e.V

erif

y th

at e

ach

eq

uat

ion

is

an i

den

tity

.

1.�c so ins

AA�

�� ta

n1A

�2.

�t sa innBB

��

� co1 s

B�

�c so insAA

��

�b c�

�a c��

�b a��

� tan1

A�

�t sa innBB

��

�b a�

�b c��

�c a��

� co1 s

B�

3.ta

n B

cos

B�

sin

B4.

1�

(cos

B)2

�(s

in B

)2

tan

B c

os

B�

�b a�

�a c��

�b c��

sin

B1(

cos

B)2

�1

��a c� 2

��c c2 2�

��a c2 2�

��c2

c�2

a2�

��b c2

2 �o

r(s

in B

)2

Try

sev

eral

val

ues

for

x t

o te

st w

het

her

eac

h e

qu

atio

n c

ould

be

an i

den

tity

.

5.co

s 2x

�(c

os x

)2�

(sin

x)2

6.co

s (9

0�

x)�

sin

x

Yes;

see

stu

den

ts’w

ork

.Ye

s;se

e st

ud

ents

’wo

rk.

B

AC

ca

b

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-6

7-6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Th

e L

aw o

f C

osi

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-7

7-7

©G

lenc

oe/M

cGra

w-H

ill38

7G

lenc

oe G

eom

etry

Lesson 7-7

The

Law

of

Co

sin

esA

not

her

rel

atio

nsh

ip b

etw

een

th

e si

des

and

angl

es o

f an

y tr

ian

gle

is c

alle

d th

e L

aw o

f C

osin

es.Y

ou c

an u

se t

he

Law

of

Cos

ines

if

you

kn

ow t

hre

e si

des

of a

tria

ngl

e or

if

you

kn

ow t

wo

side

s an

d th

e in

clu

ded

angl

e of

a t

rian

gle.

Let

�A

BC

be a

ny t

riang

le w

ith a

, b,

and

cre

pres

entin

g th

e m

easu

res

of t

he s

ides

opp

osite

L

aw o

f C

osi

nes

the

angl

es w

ith m

easu

res

A,

B,

and

C,

resp

ectiv

ely.

The

n th

e fo

llow

ing

equa

tions

are

tru

e.

a2

�b

2�

c2�

2bc

cos

Ab

2�

a2

�c2

�2a

cco

s B

c2�

a2

�b

2�

2ab

cos

C

In �

AB

C,f

ind

c.

c2�

a2�

b2�

2ab

cos

CLa

w o

f C

osin

es

c2�

122

�10

2�

2(12

)(10

)cos

48°

a�

12,

b�

10,

m�

C�

48

c�

�12

2�

�10

2�

�2(

12)

�(1

0)co

�s

48°

�Ta

ke t

he s

quar

e ro

ot o

f ea

ch s

ide.

c�

9.1

Use

a c

alcu

lato

r.

In �

AB

C,f

ind

m�

A.

a2�

b2�

c2�

2bc

cos

ALa

w o

f C

osin

es

72�

52�

82�

2(5)

(8)

cos

Aa

�7,

b�

5, c

�8

49 �

25 �

64 �

80 c

os A

Mul

tiply

.

�40

��

80 c

os A

Sub

trac

t 89

fro

m e

ach

side

.

�1 2��

cos

AD

ivid

e ea

ch s

ide

by �

80.

cos�

1�1 2�

�A

Use

the

inve

rse

cosi

ne.

60°

�A

Use

a c

alcu

lato

r.

Fin

d e

ach

mea

sure

usi

ng

the

give

n m

easu

res

from

�A

BC

.Rou

nd

an

gle

mea

sure

sto

th

e n

eare

st d

egre

e an

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

1.If

b�

14,c

�12

,an

d m

�A

�62

,fin

d a.

13.5

2.If

a�

11,b

�10

,an

d c

�12

,fin

d m

�B

.51

3.If

a�

24,b

�18

,an

d c

�16

,fin

d m

�C

.42

4.If

a�

20,c

�25

,an

d m

�B

�82

,fin

d b.

29.8

5.If

b�

18,c

�28

,an

d m

�A

�59

,fin

d a.

24.3

6.If

a�

15,b

�19

,an

d c

�15

,fin

d m

�C

.51

58

7C

B

A

48�

1210

c

C

BA

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill38

8G

lenc

oe G

eom

etry

Use

th

e La

w o

f C

osi

nes

to

So

lve

Pro

ble

ms

You

can

use

th

e L

aw o

f C

osin

esto

solv

e so

me

prob

lem

s in

volv

ing

tria

ngl

es.

Let

�A

BC

be a

ny t

riang

le w

ith a

, b,

and

cre

pres

entin

g th

e m

easu

res

of t

he s

ides

opp

osite

the

L

aw o

f C

osi

nes

angl

es w

ith m

easu

res

A,

B,

and

C,

resp

ectiv

ely.

The

n th

e fo

llow

ing

equa

tions

are

tru

e.

a2

�b

2�

c2�

2bc

cos

Ab

2�

a2

�c2

�2a

cco

s B

c2�

a2

�b

2�

2ab

cos

C

Ms.

Jon

es w

ants

to

pu

rch

ase

a p

iece

of

lan

d w

ith

th

e sh

ape

show

n.F

ind

th

e p

erim

eter

of

the

pro

per

ty.

Use

th

e L

aw o

f C

osin

es t

o fi

nd

the

valu

e of

a.

a2�

b2�

c2�

2bc

cos

ALa

w o

f C

osin

es

a2�

3002

�20

02�

2(30

0)(2

00)

cos

88°

b�

300,

c�

200,

m�

A�

88

a�

�13

0,0

�00

��

120,

0�

00 c

os�

88°

�Ta

ke t

he s

quar

e ro

ot o

f ea

ch s

ide.

�35

4.7

Use

a c

alcu

lato

r.

Use

th

e L

aw o

f C

osin

es a

gain

to

fin

d th

e va

lue

of c

.

c2�

a2�

b2�

2ab

cos

CLa

w o

f C

osin

es

c2�

354.

72�

3002

�2(

354.

7)(3

00)

cos

80°

a�

354.

7, b

�30

0, m

�C

�80

c�

�21

5,8

�12

.09

��

21�

2,82

0�

cos

8�

0°�Ta

ke t

he s

quar

e ro

ot o

f ea

ch s

ide.

�42

2.9

Use

a c

alcu

lato

r.

Th

e pe

rim

eter

of

the

lan

d is

300

�20

0 �

422.

9 �

200

or a

bou

t 12

23 f

eet.

Dra

w a

fig

ure

or

dia

gram

to

go w

ith

eac

h e

xerc

ise

and

mar

k i

t w

ith

th

e gi

ven

info

rmat

ion

.Th

en s

olve

th

e p

rob

lem

.Rou

nd

an

gle

mea

sure

s to

th

e n

eare

st d

egre

ean

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

1.A

tri

angu

lar

gard

en h

as d

imen

sion

s 54

fee

t,48

fee

t,an

d 62

fee

t.F

ind

the

angl

es a

t ea

chco

rner

of

the

gard

en.

75°;

48°;

57°

2.A

par

alle

logr

am h

as a

68°

angl

e an

d si

des

8 an

d 12

.Fin

d th

e le

ngt

hs

of t

he

diag

onal

s.11

.7;

16.7

3.A

n a

irpl

ane

is s

igh

ted

from

tw

o lo

cati

ons,

and

its

posi

tion

for

ms

an a

cute

tri

angl

e w

ith

them

.Th

e di

stan

ce t

o th

e ai

rpla

ne

is 2

0 m

iles

fro

m o

ne

loca

tion

wit

h a

n a

ngl

e of

elev

atio

n 4

8.0°

,an

d 40

mil

es f

rom

th

e ot

her

loc

atio

n w

ith

an

an

gle

of e

leva

tion

of

21.8

°.H

ow f

ar a

part

are

th

e tw

o lo

cati

ons?

50.5

mi

4.A

ran

ger

tow

er a

t po

int

Ais

dir

ectl

y n

orth

of

a ra

nge

r to

wer

at

poin

t B

.A f

ire

at p

oin

t C

is o

bser

ved

from

bot

h t

ower

s.T

he

dist

ance

fro

m t

he

fire

to

tow

er A

is 6

0 m

iles

,an

d th

edi

stan

ce f

rom

th

e fi

re t

o to

wer

Bis

50

mil

es.I

f m

�A

CB

�62

,fin

d th

e di

stan

ce b

etw

een

the

tow

ers.

57.3

mi

200

ft

300

ft

300

ft

88�80

�c

a

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Th

e L

aw o

f C

osi

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-7

7-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Th

e L

aw o

f C

osi

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-7

7-7

©G

lenc

oe/M

cGra

w-H

ill38

9G

lenc

oe G

eom

etry

Lesson 7-7

In �

RS

T,g

iven

th

e fo

llow

ing

mea

sure

s,fi

nd

th

e m

easu

re o

f th

e m

issi

ng

sid

e.

1.r

�5,

s�

8,m

�T

�39

t�

5.2

2.r

�6,

t�

11,m

�S

�87

s�

12.3

3.r

�9,

t�

15,m

�S

�10

3s

�19

.2

4.s

�12

,t�

10,m

�R

�58

r�

10.8

In �

HIJ

,giv

en t

he

len

gth

s of

th

e si

des

,fin

d t

he

mea

sure

of

the

stat

ed a

ngl

e to

th

en

eare

st t

enth

.

5.h

�12

,i�

18,j

�7;

m�

H24

.7

6.h

�15

,i�

16,j

�22

;m�

I46

.7

7.h

�23

,i�

27,j

�29

;m�

J70

.4

8.h

�37

,i�

21,j

�30

;m�

H91

.3

Det

erm

ine

wh

eth

er t

he

Law

of

Sin

esor

th

e L

aw o

f C

osin

essh

ould

be

use

d f

irst

to

solv

e ea

ch t

rian

gle.

Th

en s

olve

eac

h t

rian

gle.

Rou

nd

an

gle

mea

sure

s to

th

e n

eare

std

egre

e an

d s

ide

mea

sure

s to

th

e n

eare

st t

enth

.

9.10

.

Co

sin

es;

m�

A �

34;

Sin

es;

m�

L�

67;

m�

B�

80;

c�

30.7

m�

N�

27;

��

47.8

11.a

�10

,b�

14,c

�19

12.a

�12

,b�

10,m

�C

�27

Co

sin

es;

m�

A�

31;

Co

sin

es;

m�

A�

97;

m�

B�

46;

m�

C�

103

m�

B�

56;

c�

5.5

Sol

ve e

ach

�R

ST

des

crib

ed b

elow

.Rou

nd

mea

sure

s to

th

e n

eare

st t

enth

.

13.r

�12

,s�

32,t

�34

m�

R�

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ing

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103.

7

Det

erm

ine

wh

eth

er t

he

Law

of

Sin

es o

r th

e L

aw o

f C

osin

es s

hou

ld b

e u

sed

fir

st t

oso

lve

�A

BC

.Th

en s

ole

each

tri

angl

e.R

oun

d a

ngl

e m

easu

res

to t

he

nea

rest

deg

ree

and

sid

e m

easu

re t

o th

e n

eare

st t

enth

.

9.a

�13

,b�

18,c

�19

10.a

�6,

b�

19,m

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sin

es;

m�

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41;

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sin

es;

m�

A�

15;

m�

B�

65;

m�

C�

74m

�B

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7;c

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.7

11.a

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72

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es;

m�

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sin

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m�

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48;

m�

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95;

c�

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Sol

ve e

ach

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crib

ed b

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nd

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e n

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30.4

17.R

EAL

ESTA

TET

he

Esp

osit

o fa

mil

y pu

rch

ased

a t

rian

gula

r pl

ot o

f la

nd

on w

hic

h t

hey

plan

to

buil

d a

barn

an

d co

rral

.Th

e le

ngt

hs

of t

he

side

s of

th

e pl

ot a

re 3

20 f

eet,

286

feet

,an

d 30

5 fe

et.W

hat

are

th

e m

easu

res

of t

he

angl

es f

orm

ed o

n e

ach

sid

e of

th

e pr

oper

ty?

65.5

,54.

4,60

.1

Pra

ctic

e (

Ave

rag

e)

Th

e L

aw o

f C

osi

nes

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-7

7-7



Readin

g t

o L

earn

Math

em

ati

csT

he

Law

of

Co

sin

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-7

7-7

©G

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1G

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eom

etry

Lesson 7-7

Pre-

Act

ivit

yH

ow a

re t

rian

gles

use

d i

n b

uil

din

g d

esig

n?

Rea

d th

e in

trod

uct

ion

to

Les

son

7-7

at

the

top

of p

age

385

in y

our

text

book

.

Wh

at c

ould

be

a di

sadv

anta

ge o

f a

tria

ngu

lar

room

?S

amp

le a

nsw

er:

Fu

rnit

ure

will

no

t fi

t in

th

e co

rner

s.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.A

ccor

din

g to

th

e L

aw o

f C

osin

es,w

hic

h

stat

emen

ts a

re c

orre

ct f

or �

DE

F?

B,E

,HA

.d

2�

e2�

f2�

efco

s D

B.e

2�

d2

�f2

�2d

fco

s E

C.

d2

�e2

�f2

�2e

fco

s D

D.f

2�

d2

�e2

�2e

fco

s F

E.

f2�

d2

�e2

�2d

eco

s F

F.d

2�

e2�

f2

G.�

sin d

D ��

�sin e

E ��

�sin f

F�

H.d

��

e2�

f�

2�

2e�

fco

s �

D�

2.E

ach

of

the

foll

owin

g de

scri

bes

thre

e gi

ven

par

ts o

f a

tria

ngl

e.In

eac

h c

ase,

indi

cate

wh

eth

er y

ou w

ould

use

th

e L

aw o

f S

ines

or

the

Law

of

Cos

ines

fir

st i

n s

olvi

ng

a tr

ian

gle

wit

h t

hos

e gi

ven

par

ts.(

In s

ome

case

s,on

ly o

ne

of t

he

two

law

s w

ould

be

use

d in

sol

vin

gth

e tr

ian

gle.

)

a.S

SS

Law

of

Co

sin

esb

.AS

A L

aw o

f S

ines

c.A

AS

Law

of

Sin

esd

.SA

S L

aw o

f C

osi

nes

e.S

SA

Law

of

Sin

es

3.In

dica

te w

het

her

eac

h s

tate

men

t is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,exp

lain

wh

y.

a.T

he

Law

of

Cos

ines

app

lies

to

righ

t tr

ian

gles

. tru

eb

.T

he

Pyt

hag

orea

n T

heo

rem

app

lies

to

acu

te t

rian

gles

.Fal

se;

sam

ple

an

swer

:It

on

ly a

pp

lies

to r

igh

t tr

ian

gle

s.c.

Th

e L

aw o

f C

osin

es i

s u

sed

to f

ind

the

thir

d si

de o

f a

tria

ngl

e w

hen

you

are

giv

en t

he

mea

sure

s of

tw

o si

des

and

the

non

incl

ude

d an

gle.

Fals

e;sa

mp

le a

nsw

er:

It is

use

d w

hen

yo

u a

re g

iven

th

e m

easu

res

of

two

sid

es a

nd

th

e in

clu

ded

ang

le.

d.

Th

e L

aw o

f C

osin

es c

an b

e u

sed

to s

olve

a t

rian

gle

in w

hic

h t

he

mea

sure

s of

th

e th

ree

side

s ar

e 5

cen

tim

eter

s,8

cen

tim

eter

s,an

d 15

cen

tim

eter

s.Fa

lse;

sam

ple

answ

er:

5 �

8 �

15,s

o,b

y th

e Tr

ian

gle

Ineq

ual

ity

Th

eore

m,n

o s

uch

tria

ng

le e

xist

s.

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al f

orm

ula

is

to r

elat

e it

to

one

you

alr

eady

know

.Th

e L

aw o

f C

osin

es l

ooks

som

ewh

at l

ike

the

Pyt

hag

orea

n T

heo

rem

.Bot

h f

orm

ula

sm

ust

be

tru

e fo

r a

righ

t tr

ian

gle.

How

can

th

at b

e? c

os

90 �

0,so

in a

rig

ht

tria

ng

le,w

her

e th

e in

clu

ded

an

gle

is t

he

rig

ht

ang

le,t

he

Law

of

Co

sin

esb

eco

mes

th

e P

yth

ago

rean

Th

eore

m.

D

dE

e F

f

©G

lenc

oe/M

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ill39

2G

lenc

oe G

eom

etry

Sp

her

ical

Tri

ang

les

Sph

eric

al t

rigo

nom

etry

is

an e

xten

sion

of

plan

e tr

igon

omet

ry.

Fig

ure

s ar

e dr

awn

on

th

e su

rfac

e of

a s

pher

e.A

rcs

of g

reat

ci

rcle

s co

rres

pon

d to

lin

e se

gmen

ts i

n t

he

plan

e.T

he

arcs

of

thre

e gr

eat

circ

les

inte

rsec

tin

g on

a s

pher

e fo

rm a

sph

eric

al

tria

ngl

e.A

ngl

es h

ave

the

sam

e m

easu

re a

s th

e ta

nge

nt

lin

es

draw

n t

o ea

ch g

reat

cir

cle

at t

he

vert

ex.

Sin

ce t

he

side

s ar

e ar

cs,t

hey

too

can

be

mea

sure

d in

deg

rees

.

Sol

ve t

he

sph

eric

al t

rian

gle

give

n a

�72

�,b

�10

5�,a

nd

c�

61�.

Use

th

e L

aw o

f C

osin

es.

0.30

90�

(–0

.258

8)(0

.484

8)�

(0.9

659)

(0.8

746)

cos

Aco

s A

�0.

5143

A�

59°

�0.

2588

� (

0.30

90)(

0.48

48)�

(0.9

511)

(0.8

746)

cos

Bco

s B

��

0.49

12B

�11

9°

0.48

48�

(0.

3090

)(–0

.258

8)�

(0.9

511)

(0.9

659)

cos

Cco

s C

�0.

6148

C�

52°

Ch

eck

by u

sin

g th

e L

aw o

f S

ines

.

�s si in n7 52 9° °�

��s si in n

1 10 15 9° °�

��s si in n

6 51 2° °�

�1.

1

Sol

ve e

ach

sp

her

ical

tri

angl

e.

1.a

�56

°,b

�53

°,c

�94

°2.

a�

110°

,b�

33°,

c�

97°

A�

41�,

B�

39�,

C�

128�

A�

116�

,B�

31�,

C�

71�

3.a

�76

°,b

�11

0°,C

�49

°4.

b�

94°,

c�

55°,

A�

48°

A�

59�,

B�

124�

,c�

59�

a�

60�,

B�

121�

,C�

45�

A

C

B

c

ba

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

7-7

7-7

The

sum

of

the

side

s of

a s

pher

ical

tria

ngle

is le

ss t

han

360°

.T

he s

um o

f th

e an

gles

is g

reat

er t

han

180°

and

less

tha

n 54

0°.

The

Law

of

Sin

es f

or s

pher

ical

tria

ngle

s is

as

follo

ws.

� ss ii nnAa

��

� ss ii nnBb

��

� ss ii nnCc

�

The

re is

als

o a

Law

of

Cos

ines

for

sph

eric

al t

riang

les.

cos

a�

cos

bco

s c

�si

n b

sin

cco

s A

cos

b�

cos

aco

s c

�si

n a

sin

cco

s B

cos

c�

cos

aco

s b

�si

n a

sin

bco

s C

Exam

ple

Exam

ple


chapter 7 resource masters - math problem...

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