Honors Geometry 9 March 2012WarmWarm Up Up

1) Find the volume of a hemisphere if r = 6Note: Vsphere =

2) How do length, area and volume formulas differ?

34

3r

ObjectiveStudents will develop strategies for solving problems

involving volume.

Students will work on a problem independently and evaluate work samples with their group.

DUE TODAYTasks 3 & 4 of Coordinate Geometry Project

REMINDER- all necessary formulas are in your book Task 7: DISCUSS translations…

projects Coordinate Geometry Project DUE March 13th

REVISIONS for DSH Kribz, due MARCH 13

Honors Geometry 4th Quarter Project DETAILS- PROPOSAL APPLICATION- ONLINE Preliminaries– due March 13 Final project due- May 8th

video, song, skit, tutorial, rap, dance…. other?

Work independently / silentlyconsider my comments….. work for 10 minutes…..

Read the questions and try to answer them as carefully as you can. Show all your work so I can understand your reasoning. Use another paper if you need more room. In addition to solving the task, I want to see if you can present your work in an organized and clear manner.

3r

debrief

what math did we do today?

what was easy?

what is still confusing?

Finding the Distance Between Two Points

Using the Pythagorean theorem

(x 2 – x 1) 2 + ( y 2 – y 1) 2 = d 2

THE DISTANCE FORMULA

The distance d between the points (x 1, y 1) and (x 2, y 2) is

d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

Solving this for d produces the

distance formula.

You can write the equation

a 2 + b 2 = c 2

x 2 – x 1

y2 – y1

d

x

y

C (x 2, y 1 )

B (x 2, y 2 )

A (x 1, y 1 )

The steps used in the investigation can be used to develop a general formula for the distance between two points A(x 1, y 1) and B(x 2, y 2).

Finding the Distance Between Two Points

Find the distance between (1, 4) and (–2, 3).

d = (x 2 – x 1) 2 + ( y 2 – y 1) 2

= 10

3.16

To find the distance, use the distance formula.

Write the distance formula.

Substitute.

Do the math

SOLUTION

= (x 2 – x 1) 2 + ( y 2 – y 1) 2–2 – 1 3 – 4

Applying the Distance Formula

A player kicks a soccer ball that is 10 yards from a sideline and 5 yards from a

goal line. The ball lands 45 yards from the same goal line and 40 yards from

the same sideline. How far was the ball kicked?

The ball is kicked from the point (10, 5),

and lands at the point (40, 45). Use the

distance formula.

d = (40 – 10) 2 + (45 – 5) 2

= 900 + 1600 = 2500 = 50

The ball was kicked 50 yards.

SOLUTION

Finding the Midpoint Between Two Points

The midpoint of a line segment is the point on the segment that is equidistant

from its end-points. The midpoint between two points is the midpoint of the line

segment connecting them.

THE MIDPOINT FORMULA

The midpoint between the points (x 1, y 1) and (x 2, y 2) isx 1 + x 2

2( )y 1 + y 2

2,

Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check

the result.

SOLUTION

–2 + 42( )3 + 2

2,22( )5

2,= 1( )52,=

The midpoint is , . 1( )52

x 1 + x 2

2( )y 1 + y 2

2,Remember, the midpoint formula is .

Finding the Midpoint Between Two Points

Find the midpoint between the points (–2, 3) and (4, 2). Use a graph to check

the result.

CHECK

From the graph, you can see that the

point , appears halfway

between (–2, 3) and (4, 2). You can also

use the distance formula to check that

the distances from the midpoint to each

given point are equal.

( )152

Finding the Midpoint Between Two Points

(1, )52

(–2, 3)

(4, 2)

Applying the Midpoint Formula

You are using computer software to design a video game. You want to place

a buried treasure chest halfway between the center of the base of a palm

tree and the corner of a large boulder. Find where you should place the

treasure chest.

SOLUTION

Assign coordinates to the locations of the two landmarks. The center of the palm tree is at (200, 75). The corner of the boulder is at (25, 175).

Use the midpoint formula to find the point that is halfway between the two landmarks.

1

2

25 + 2002( )175 + 75

2,225

2( )2502,= = (112.5, 125)

(25, 175)

(200, 75)

(112.5, 125)

Term Definition Example

Distance Formula

The distance between points and Is given by the formula:

Chapter 9 Pythagorean Theorem

2 2 2a b c

1 1,A x y 1, 2 , 11, 7

2 2,B x y

2 2

2 1 2 1AB x x y y

Find the distance between each pair of Find the distance between each pair of pointspoints 1) 1, 2 , 11, 7 2) 9, 6 , 3,10

practiceDo pg. 504: 1 – 3

groups- DO ON YOUR OWN GRAPH PAPER THEN PREPARE A POSTER TO SHOW YOUR WORKGroups 1 and 7: #7Groups 2 and 6: #8Groups 3 and 5: #9Groups 4 and 8: #10

debrief

how did we use Pythagorean formula to develop the distance formula?

what is easy?

what is still confusing?

CW: Do problems page 548THINK– SILENTLY make a sketch write basic information on your paper write formulas you can use how can you find the answer?PAIR- chat with a partner- complete problemSHARE- class discussion

DO problems: # 8, 11 DO problems: # 8, 11 (1 Cubic Foot = 7.5 gallons)(1 Cubic Foot = 7.5 gallons)

# 13, 15# 13, 15BE READY TO SHARE YOUR ANSWERS in 20 minutesBE READY TO SHARE YOUR ANSWERS in 20 minutes

a

0 0 045 45 90

045

a

a2a

0 0 030 60 90

060a

2a3a

COPY IN YOUR NOTES

CW: Do pg. 493: 1 - 6

VOLUME and SA

BASICALLY,

SA = AREASA = AREAbase(s)base(s) + LSA + LSA

(add the areas of the faces)

VVPRISM/CYLINDERPRISM/CYLINDER = A = ABASEBASE x HEIGHT x HEIGHT

HEIGHT is ALWAYS PERPENDICULAR distance

VolumeVolume

Rectangular Prism

Triangular prism

2V r h

h

V l w h w

l

h

Trapezoidal prism

2

bhV H

H

1 2

2

h b bV H

HCylinder Prism

Pyramid

3

BV h

3

BV h

2( )base B r

Con

e

Sphe

re

34

3V r

24SA r

hh

( )base B l w

hb l w

V = Abase∙H

a

#1 #2

#3 #4

Geometric Probability

outcome- a possible result

event- a set of desired outcomes

probability- the chance that something will happen, expressed as a decimal, fraction or %

Probability = -----------------------------------# of desired outcomes

total # of outcomes possible

a

3 26

x

Solve for xSolve for x2

4 105

x

66 8

x

12 33

x x

Geometric Probability pg. 458The Shape of Things- go over as class

Right on Target

Work with your group to do the investigation, steps 5 - 9. Each student must submit a write up clearly showing their work for each step.

Be prepared to share your results for steps 5- 9 with the class. Be ready to justify your answer.

Area problemMr. Brown’s class will raise rabbits for their spring

science fair. They have 24 feet of fencing with which to build a rectangular pen to keep the rabbits.

1) If Mr. Brown’s students want their rabbits to have as much room as possible, how long will each of the sides of the pen be?

2) How long would each of the sides be if they had 16 feet of fencing?

3) How would you determine the pen with the most room for any amount of fencing? Organize your work so that someone else who reads it will understand it.

Midpoint on graph with variables

Midpoint =

THE AVERAGE of the coordinates

2 1 2 1( , )2 2

x x y y (x2, y2)

(x1, y1)

rise y change in yslope m

run x change in x

x y1 24 6

(4, 6)

(1, 2)

6 - 24 - 1

6 2 41.33

4 1 3m

2

b h

Area formulas:

b h 1 2

2

h b b1 2

2

d d

Regular Polygons

2

a s nA

or

2

a PA

Circles

r2A r

2C r

C dor

VOLUMEcylinder or prism =Abase H

VOLUMEcone or pyramid =⅓Abase H

(x1, y1)

(x2, y2)x yx1 y1

x2 y2

y2-y1(x2-x1)

2 1

2 1

y ym

x x

Point- Slope form of linear equation

2 1

2 1

y ym

x x

m(x2-x1)= y2 –y1

y2 –y1=m(x2-x1)

y2 = y1 + m(x2-x1)

y = y1 + m(x-x1)

Challenge QuestionImagine a steel belt fitting tightly around Earth’s

equator. Now imagine cutting the belt and splicing in a piece to make the belt 40 feet longer. Make the longer belt stand out evenly from the equator. (HINT- Cearth≈ 24901 miles)

What’s the largest object that will fit under the belt: an atom? an ant? a large dog?

an elephant?Explain your answer in complete sentences. You

may make a sketch to help you think about it.

a

steps for finding surface area1. Draw and label each face of the solid as if you had cut the

solid apart along its edges and laid it flat. Label the dimensions.

2. Calculate the area of each face: Formula

Substitute the #’s

Do the math

UnitsUnits

3. Find the total area. Basically….Basically….surface area = area of base(s) + area of sidessurface area = area of base(s) + area of sides

proof…..paragraph, two column or flow chart

How can you prove that the formula for the area of a parallelogram is A = bh?Can you PROVE that both triangles are congruent?

http://www.basic-mathematics.com/proof-of-the-area-of-a-parallelogram.html

Quiz

Work silently on your quiz. Show all steps and justify your answers (use math).

When you are finished, you may silentlybegin to work on your project.

a

2C r 2A r24cm

1. Determine the area of the circle.Diameter = 24 cm

2.

3.

4.

Practice:1.1. Write the Area formula inside the appropriate figure:Write the Area formula inside the appropriate figure:

4. A garden 4 ft by 8 4. A garden 4 ft by 8 is surrounded by a is surrounded by a

sidewalk 3 feet wide– sidewalk 3 feet wide– Determine the area of Determine the area of

the sidewalkthe sidewalk

2. A rectangle yard is 20 meters by 44 meters. 2. A rectangle yard is 20 meters by 44 meters. If a rectangular swimming pool 9 meters by 11 If a rectangular swimming pool 9 meters by 11 meters is put in the yard, how much yard area meters is put in the yard, how much yard area

is left?is left?3. The area is 64, find h3. The area is 64, find h

h

16

Do Now:

2 2 2a b c f

1.1. Find the area of each figure:Find the area of each figure:

C

4. If the perimeter is 4. If the perimeter is 38, 38,

what is the base (b)?what is the base (b)?What is the area?What is the area?

20cm x

15in

BDAB CB

2. A rectangle yard is 33 meters by 40 meters. If a 2. A rectangle yard is 33 meters by 40 meters. If a rectangular swimming pool 8 meters by 7 meters is rectangular swimming pool 8 meters by 7 meters is

put in the yard, how much yard area is left?put in the yard, how much yard area is left?

3. Find the value of x3. Find the value of x

5

b

xA

8

c 62o132o

3in

3

7 18

x

x

3 6

DEAB

12cm

4cm

5cm

15cm

a

Cla

ssw

ork

C

lass

work

We also need to know how to USE each formula and apply them to solve problems!Find the area of the shaded region Show all steps

exit quizDo your quiz- show all steps clearly.

Solve for x and check A: x2 + 3x – 4 = 0 B: x2 + 5x + 4 = 0 C: x2 + 5x – 6 = 0 D: x2 + 7x + 6 = 0

Using the number π

Keep the number π in your answer if “exact answer” is required.

Keep the number π in your calculations until the last step, to avoid rounding errors. It is best

to use the π button on your calculator for your decimal approximation. If you don’t have a π

button, you can substitute 3.14 for π.

a

#1 #2

#3 #4

a

2C r

2A r

24cm

1. Determine the area of the circle.Diameter = 24 cm

2.

3.

4.

Properties of Circles

Central Angles:An angle that has its vertex at the center of a circle

0280132

028

0132

Sector of a circle is the region between two radii and an arc of a circle.

Term Definition Example

Circle sector area

conjecture

The area of a sector of a circle is given by the formula,

A is the area and r is the radius of the circle, and ‘a’ is the degree of the

inscribed angle

2 2 2a b c

2

360

aA r

a

proof…..paragraph, two column or flow chart

How can you prove that the formula for the area of a parallelogram is A = bh?Can you PROVE that both triangles are congruent?

http://www.basic-mathematics.com/proof-of-the-area-of-a-parallelogram.html

Practice:1.1. Write the Area formula inside the appropriate figure:Write the Area formula inside the appropriate figure:

4. A garden 4 ft by 8 4. A garden 4 ft by 8 is surrounded by a is surrounded by a

sidewalk 3 feet wide– sidewalk 3 feet wide– Determine the area of Determine the area of

the sidewalkthe sidewalk

2. A rectangle yard is 20 meters by 44 meters. 2. A rectangle yard is 20 meters by 44 meters. If a rectangular swimming pool 9 meters by 11 If a rectangular swimming pool 9 meters by 11 meters is put in the yard, how much yard area meters is put in the yard, how much yard area

is left?is left?3. The area is 64, find h3. The area is 64, find h

h

16

Do Now:

2 2 2a b c f

1.1. Find the area of each figure:Find the area of each figure:

C

4. If the perimeter is 4. If the perimeter is 38, 38,

what is the base (b)?what is the base (b)?What is the area?What is the area?

20cm x

15in

BDAB CB

2. A rectangle yard is 33 meters by 40 meters. If a 2. A rectangle yard is 33 meters by 40 meters. If a rectangular swimming pool 8 meters by 7 meters is rectangular swimming pool 8 meters by 7 meters is

put in the yard, how much yard area is left?put in the yard, how much yard area is left?

3. Find the value of x3. Find the value of x

5

b

xA

8

c 62o132o

3in

3

7 18

x

x

3 6

DEAB

12cm

4cm

5cm

15cm

a

Cla

ssw

ork

C

lass

work

We also need to know how to USE each formula and apply them to solve problems!Find the area of the shaded region Show all steps

exit quizDo your quiz- show all steps clearly.

Solve for x and check A: x2 + 3x – 4 = 0 B: x2 + 5x + 4 = 0 C: x2 + 5x – 6 = 0 D: x2 + 7x + 6 = 0

ObjectiveStudents will find surface area and volume of various

geometric solids.

Students will take notes, participate in class discussion and use think-pair-share.

NEED TO TAKE TEST TODAY- P1: Titus, Bryan, Lauren H, AmmarP6: Seila

Geometric SolidsGeometric Solids 2 Bases2 Bases 1 Base1 Base No BaseNo Base

Prisms & Prisms & CylindersCylinders

Cones & Cones & pyramidspyramids

SpheresSpheres

Bases are congruent and parallel

Stu

dy

Sheet

VolumeVolume RectangularPrism

Triangular prism

2base r

Hbase l w

wl

H

Trapezoidal prism

1

2base bh

H

1 2

1

2base h b b

HCylinder Prism

Pyramid

2base r

Con

e

V = AV = Abasebase∙ H∙ H

V = 1/3 Abase∙ H V = 1/3 Abase∙ H