Looking Ahead to Chapter 7 - Michele Clark

Looking Ahead to Chapter 7Focus In Chapter 7, you will learn how to identify and find unknown measures in similar

polygons and solids, prove that two triangles are similar, and use indirect measurement

to solve problems.

Chapter WarmupAnswer these questions to help you review skills that you will need in Chapter 7.

Write each fraction in simplest form.

1. 2. 3.

Multiply. Write your answer in simplest form.

4. 5. 6.

Solve each proportion.

7. 8. 9.

Read the problem scenario below.

You look up into the sky and see an airplane flying toward the airport. The airplane is 22,176

feet directly above you. You are 55 miles from the airport.

10. How far, in miles, are you from the airplane?

11. About how far, in miles, is the airplane from the airport?

4

50�

x

200

8

20�

6

x

x

8�

21

24

5

2�

7

25

9

2�

8

27

1

4�

5

6

56

68

15

65

18

12

280 Chapter 7 ■ Similarity

© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

ratio ■ p. 284

proportion ■ p. 287

means ■ p. 287

extremes ■ p. 287

similar ■ p. 290

congruent ■ p. 290

scale model ■ p. 295

scale ■ p. 295

paragraph proof ■ p. 304

indirect measurement ■ p. 305

similar solids ■ p. 312

scale factor ■ p. 312

Key Terms

Chapter 7 ■ Similarity 281

© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Similarity

C H A P T E R

7

The first plastic containers for storing food were introduced in 1946. These containers

grew in popularity because they were more durable than glassware and kept food

fresh longer. In Lesson 7.5, you will compare the surface area and volume of similar

plastic containers.

7.1 Ace Reporter

Ratio and Proportion ■ p. 283

7.2 Framing a Picture

Similar and Congruent Polygons ■ p. 289

7.3 Using an Art Projector

Proving Triangles Similar: AA, SSS,

and SAS ■ p. 297

7.4 Modeling a Park

Indirect Measurement ■ p. 305

7.5 Making Plastic Containers

Similar Solids ■ p. 311


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Lesson 7.1 ■ Ratio and Proportion 283

© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Ace ReporterRatio and Proportion

ObjectivesIn this lesson,

you will:

■ Write and simplify

ratios.

■ Compare ratios.

■ Write and solve

proportions.

■ Use survey results to

make predictions.

Key Terms■ ratio

■ proportion

■ means

■ extremes

SCENARIO You are a reporter for your school’s newspaper.

You are writing an article about the order of classes during the school

day and you are interviewing students to see what they think.

Problem 1 Survey Says

From the investigating you have done so far, it seems that the

students have a strong opinion on when the physical education

class should occur. You have surveyed many students and

recorded the results in the table below.

A. How many students did you survey? Show all your work and use

a complete sentence in your answer.

B. What can you conclude from your survey results? Explain your

reasoning. Use complete sentences in your answer.

7.1

When Do You Think Gym Classes Should Be Held?

Beginning of Day End of Day Any Time

8 14 2

Investigate Problem 11. In your article you want to compare the results in your survey.

One way you could compare the results is by writing the

statement, “Eight out of 24 students prefer to have gym class

at the beginning of the day.”

Which two numbers from the survey results are being compared?

Use a complete sentence in your answer.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 1Complete the following statements that compare the numbers in

the survey.

“____________ out of 24 students prefer to have gym class at the

end of the day.”

“Two out of 24 students ___________________________________

______________________________.”

2. Just the Math: Ratio You can mathematically compare

the results in the table by using ratios. A ratio is a comparison

of two numbers that uses division. You can write a ratio as

a fraction or by using a colon. For instance, you can write

“Eight out of 24 students prefer to have gym class at the

beginning of the day” in two ways.

As a fraction:

Using a colon: 8 students : 24 students

When you use a colon, you read the colon as the word “to.”

So, the statement “8 students : 24 students” is read as

“8 students to 24 students.”

Write each of the other statements from Question 1 as a ratio.

Write each ratio as a fraction. If possible, simplify your fractions.

3. Suppose that you have only surveyed students in your own

grade. A friend of yours offers to help you out and surveys

students from another grade in your school. Your friend’s

results are shown in the table below.

How many students did your friend survey?

Write a ratio that compares the number of students that prefer

gym class at the beginning of the day to the number of students

surveyed. Write your answer as a fraction in simplest form.

8 students

24 students



15 18 3


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 1Write a ratio that compares the number of students that prefer

gym class at the end of the day to the number of students

surveyed. Write your answer as a fraction in simplest form.

Write a ratio that compares the number of students that do not

care when gym class is held to the number of students surveyed.

Write your answer as a fraction in simplest form.

4. Do a larger portion of the students in your survey or your friend’s

survey prefer to have gym class at the beginning of the day?

Show all your work and explain your reasoning. Use complete

sentences in your answer.

Do a larger portion of the students in your survey or your friend’s

survey prefer to have gym class at the end of the day? Show all

your work and explain your reasoning. Use complete sentences

in your answer.

Do a larger portion of the students in your survey or your friend’s

survey have no preference for when gym class is held? Show all

your work and explain your reasoning. Use complete sentences

in your answer.

5. When are two different ratios equivalent? Use complete

sentences to explain your reasoning.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 16. Complete the table below to show the results of your survey and

your friend’s survey together. Then write two equivalent ratios for

each statement. Write your ratios as fractions.

Students who prefer gym class at end of day : Students who

prefer gym class at beginning of day

Students who have no preference : Students who prefer gym

class at end of day



Problem 2 Making Predictions

A. Use the combined results of the surveys in Problem 1 to write the

following ratios. Write each ratio as a fraction in simplest form.

Students who prefer gym class at beginning of day : All students

surveyed

Students who prefer gym class at end of day : All students

surveyed

Students with no preference : All students surveyed


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Problem 2 Making Predictions

B. Suppose that you want to interview students from the other

grades in your school. Would you expect that the results

you would get from surveying the other grades would be very

different from the results you already have? Why or why not?

Use complete sentences in your answer.

C. Suppose that you interviewed 30 students in a different grade.

How many students would you expect to respond that they

prefer to have gym class at the end of the day? Explain your


Investigate Problem 21. Just the Math: Proportion When two ratios that compare

the same quantities are equal, you can write them as a proportion.

A proportion is an equation that states that two ratios are

equivalent, or equal. We write a proportion by placing an equals

sign between two equivalent ratios or by using a double colon in

place of the equals sign. For instance, you could have used a

proportion to answer part (C):

.

What is the value of the unknown quantity in the proportion

above? Use complete sentences to explain how you found

your answer.

When you found the unknown quantity, you were solving

the proportion.

2. Another way to solve a proportion is by using the proportion’s

means and extremes.

extremes meansa

b�

c

d

8 students

15 students�

? students

30 students


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 2What are the means of the solved proportion in Question 1?

What are the extremes of the solved proportion in Question 1?


Find the product of the means and the product of the extremes

from Question 1. What do you notice? Show all your work and

use a complete sentence in your answer.

Use the results in Question 2 to complete the steps to solve

the following proportion. Show all your work.

Set product of extremes equal to product

of means.

Divide each side by 4.

Simplify.

Use complete sentences to explain how to solve a proportion by

using the proportion’s means and extremes.

3. Suppose that there are 480 students in your school. Use the com-

bined survey results from Problem 1 to predict how many students

in your school would prefer to have gym class at the beginning of

the day, how many students would prefer to have gym class at the

end of the day, and how many students have no preference. Show

all your work and use complete sentences in your answer.

x � �

�

4�

18

�

� � �

4

3�

6

x

Lesson 7.2 ■ Similar and Congruent Polygons 289

© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Framing a PictureSimilar and Congruent Polygons


you will:

■ Identify similar and

congruent polygons.

■ Identify corresponding

angles and correspond-

ing sides in similar and

congruent polygons.

■ Find unknown measures

in similar and congruent

polygons.

■ Find unknown measures

in a scale model.

Key Terms■ similar

■ congruent

■ scale model

■ scale

SCENARIO When you frame a picture, it is not unusual to put

a mat inside the frame. A mat is a piece of paperboard that is used

to provide a transition between a picture and the picture frame.

Problem 1 The Perfect Picture

You are creating your own collage of pictures. You have bought a

large frame and will cut out rectangular holes in the mat as shown.

A. What are the interior angle measures of each mat opening?


B. Write a ratio that compares the length of rectangle A to the length

of rectangle B. Then write a ratio that compares the width of

rectangle A to the width of rectangle B. Write your answers as

fractions in simplest form.

What do you notice? Use a complete sentence in your answer.

A

BCD

4 in.5 in.

6 in. 7 in.

3 in.2 in.

2 in.3 in.

frame

mat

picture

7.2

Take NoteIn this lesson, the length

refers to the longer side of

the rectangle.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 11. Two polygons are similar when the corresponding angles are

congruent and the ratios of the measures of the corresponding

sides are equal. Which rectangles from Problem 1 are similar?

Explain your reasoning. Use complete sentences in your answer.

2. Two polygons are congruent when the corresponding angles

are congruent and the corresponding sides are congruent.

Which rectangles from Problem 1 are congruent? Explain

your reasoning. Use complete sentences in your answer.

Problem 1 The Perfect Picture

C. Write a ratio that compares the length of rectangle A to the

length of rectangle D. Then write a ratio that compares the width

of rectangle A to the width of rectangle D. Write your answers as

fractions in simplest form.


D. Write a ratio that compares the length of rectangle A to the

length of rectangle C. Then write a ratio that compares the width

of rectangle A to the width of rectangle C.


Take NoteIn a rectangle, one pair of

corresponding sides are the

lengths and the other pair

of corresponding sides are

the widths.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 13. What do you notice about the ratios of the corresponding sides

of congruent figures? Use a complete sentence in your answer.

4. Are all similar figures also congruent figures? If so, explain your

reasoning. If not, give an example that shows a pair of similar

figures that are not congruent. Use complete sentences in

your answer.

5. Are all congruent figures also similar figures? If so, explain

your reasoning. If not, give an example that shows a pair of

congruent figures that are not similar. Use complete sentences

in your answer.

6. The triangles shown below are congruent.

You can write . Whenever you write a

congruence statement like this, the letters that name the

vertices should be written in corresponding order. For instance,

so and are in the same position. Name the

pairs of corresponding angles and corresponding sides.

The measure of is 40º; the measure of is 88º; the length

of is 2.3 centimeters; and the length of is 2.8 centimeters.

Label this information on the figures above.

What is the length of ? Explain how you found your answer.


What is the measure of ? Explain how you found your

answer. Use a complete sentence in your answer.

�B

DE

DFAB

�E�A

�D�A�A � �D,

�ABC � �DEF

D

E

FA

B

C


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 17. The two triangles below are similar. You can write

where the symbol ~ means “is similar to.”

Again, the order in which you write the vertices in a similarity

statement indicates the corresponding angles and the

corresponding sides. List the corresponding angles and

the corresponding sides.

Write a ratio that compares a side length of to a

corresponding side length of .

Then write a ratio that compares a side length of to a

corresponding side length of .

Are the two ratios equal? Why or why not? Use complete


Because the triangles are similar, we can write a proportion

that relates the ratios of the lengths of the sides. One possible

proportion is

Another possible proportion is

When you write a proportion relating the corresponding side

lengths of two similar polygons, what must be true about both of

the ratios? Use complete sentences in your answer.

XZ

UW�

YZ

VW.

UV

XY�

VW

YZ.

�UVW

�XYZ

�XYZ

�UVW

X

Y

Z

U

V

W

�UVW ~ �XYZ,


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 18. In the figure below,

Complete the following proportions that relate the ratios of the

lengths of the sides of the triangle.

Suppose that 3 feet, 9 feet, and 5 feet.

Write a proportion that you can use to find LM. Then solve the

proportion. Show all your work and use a complete sentence

in your answer.

Suppose that you also know that 12 feet. Find GI.

Show all your work and use a complete sentence in your answer.

KM �

HI �KL �GH �

�LM

�GI

KM

KM

GI�

LM

�GH

KL�

HI

�

L

K M

H

G I

�GHI ~ �KLM.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 1Find the ratio of the height of to the height of .

Find the ratio of the length of the base of to the length

of the base of .

Compare the ratios of the lengths and heights. Use a complete

sentence in your answer.

Find the areas of the triangles. Then find the ratio of the area

of to the area of . Write your ratio as a fraction in

simplest form.

How is the ratio of the areas related to the ratio of the heights?

How is the ratio of the areas related to the ratio of the lengths?

Why do you think this is so? Explain your reasoning.


�KLM�GHI

�KLM

�GHI

�KLM�GHI


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 19. A scale model (or model) of an object is similar to the actual

object but is either larger or smaller. The ratio of a dimension

of the actual object to a corresponding dimension in the model

is called the scale of the model.

A wall mural is being created from a picture that is 6 inches long

and 4 inches wide. The wall mural should be 48 inches long.

Complete the statement below to find the scale of the model.

Write your answer as a fraction in simplest form.

Now use the scale to complete the proportion that you can use

to find the width of the mural.

Find the width of the mural. Show all your work and use a

complete sentence in your answer.

10. A scale model of a framed picture is being created for a

dollhouse. The actual rectangular picture is 4 inches wide and

8 inches long. The scale of the model is 4 : 1. Find the length

and width of the dollhouse picture. Show all your work and


width of picture

width of mural�

length of picture

length of mural�


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Lesson 7.3 ■ Proving Triangles Similar: AA, SSS, and SAS 297

© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Using an Art ProjectorProving Triangles Similar: AA, SSS, and SAS


you will:

■ Use given information to

show that two triangles

are similar.

■ Complete a paragraph

proof.

Key Term■ paragraph proof

7.3

SCENARIO An art projector is a piece of equipment that

artists use to create exact copies of artwork, to enlarge artwork, or

to reduce artwork. A basic art projector uses a light bulb and a lens

within a box. The light rays from the art being copied are collected

onto a lens at a single point. Then the lens projects the image of

the art onto a screen as shown below.

If the projector is set up properly, the triangles above will be similar

polygons. You can show that these triangles are similar without

measuring all of the side lengths and all of the interior angles.

Problem 1 Angles, Angles, Angles

A. Suppose that two triangles are similar. What do you know about

the two triangles? Use a complete sentence in your answer.

B. Consider the two triangles below. Without using a ruler or

protractor, can you determine whether the triangles are similar?

Why or why not? Use complete sentences in your answer.

LightLens

ArtProjector

Image

Screen


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Problem 1 Angles, Angles, Angles

C. Use a protractor to create two different triangles that each have

two interior angles that measure 35º and 45º. Label the vertices

of your triangles.

D. What do you know about the third interior angle in each of

the triangles in part (C)? Explain how you found your answer.


E. Measure the side lengths of each triangle in part (C) to the

nearest millimeter and record these lengths below. Also record

the interior angle measures below.

F. Are the triangles similar? Explain your reasoning and use a

complete sentence in your answer.

G. When the corresponding angles of two triangles are congruent,

what can you conclude about the two triangles? Use a complete

sentence in your answer.

H. If you know that two pairs of corresponding angles are

congruent, can you conclude that the triangles are similar?

Why or why not? Use complete sentences in your answer.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 11. The result of Problem 1 is called the Angle-Angle Similarity

Theorem.

Why do you think that it is enough to know that the

corresponding angles of two triangles are congruent in order

to say that the triangles are similar? In other words, why do

you not need any information about the side lengths?


2. The triangles shown are isosceles triangles. Do you have enough

information to show that the triangles are similar? Explain your

reasoning. Use a complete sentence in your answer.

P

Q

RL

M

N

Angle-Angle Similarity TheoremIf two angles of one triangle are congruent to two angles of

another triangle, then the triangles are similar.

If and , then .�ABC ~ �DEFm�C � m�Fm�A � m�D

D

E

F

A

B

C

Take NoteIn the figure at the right, the

double arcs show that

and are congruent, but

that these angles are not

congruent to and .�D�A

�F

�C


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7Take Note

In the figure, the single hash

marks indicate that

and the double hash marks

indicate that , but

no relationship between

the corresponding sides

is known.

VW � XW

ST � TU

Investigate Problem 13. The triangles shown are isosceles triangles. Do you have enough

information to show that the triangles are similar? Explain your


W

V X

T

S U

Problem 2 Sides, Sides, Sides

A. What must be true about the sides of similar triangles?


B. In Problem 1, you found that if two pairs of corresponding angles

of two triangles are congruent, then the triangles are similar.

Suppose that you drew two triangles so that the ratios of two

pairs of corresponding sides are equal. Do you think that these

triangles would necessarily be similar? Use complete sentences

to explain your reasoning.

C. Measure the lengths of the sides of the triangle below to

the nearest millimeter. Record the lengths on the triangle.

L

M

N


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 21. The result of Problem 2 is called the Side-Side-Side Similarity

Postulate.

Problem 2 Sides, Sides, Sides

What could be the side lengths of a triangle that is similar to

on the previous page?

D. Draw a triangle with the side lengths described in part (C).

Determine whether this triangle is similar to the triangle in

part (C). Explain how you found your answer. Use complete


�LMN

Side-Side-Side Similarity PostulateIf the corresponding sides of two triangles are proportional,

then the triangles are similar.

If then .�ABC ~ �DEFAB

DE�

BC

EF�

AC

DF,

D

E

F

A

B

C

Take NoteWhen it is said that the

corresponding sides are

proportional, it means that

the ratios of lengths of

the corresponding sides

are equal.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 2If the corresponding sides of two triangles are proportional,

what makes the triangles similar? Use complete sentences in

your answer.

2. Determine whether is similar to . If so, use

symbols to write a similarity statement. Show all your work

and use a complete sentence in your answer.

3. An art projector has been set up properly, and a piece of art has

been projected onto a wall, as shown below. The triangles below

are isosceles triangles.

Find the unknown side lengths of and . Show all

your work. Leave your answers as radicals in simplest form.

�PQN�LMN

N

Q

P

M

L

Image

Lens

Art4 inches 12 inches

8 inches 24 inches

16 meters

24 meters

22 meters

Z

X

Y

U

V

W33 meters

36 meters

24 meters

�XYZ�UVW


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 2Show that . Explain your reasoning.


4. Suppose that you know that the ratios of the lengths of two

pairs of corresponding sides of two triangles are equal. How

many pairs of corresponding angles do you need to know are

congruent in order to determine that the triangles are similar?

Which pair(s) of angles would these have to be? Use complete

sentences to explain your reasoning.

5. In Question 4, you should have discovered the Side-Angle-Side

Similarity Postulate.

Can you use this postulate to show that the triangles in

Question 3 are similar? If so, explain which angles you would

show are congruent and which sides you would show are

proportional. Use complete sentences in your answer.

�LMN ~ �PQN

Side-Angle-Side Similarity PostulateIf two of the corresponding sides of two triangles are

proportional and the included angles are congruent,

then the triangles are similar.

If and then .�ABC ~ �DEF�A � �D,AB

DE�

AC

DF

D

E

F

A

B

C


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 26. You can use a paragraph proof to prove that

in Question 3. A paragraph proof is a proof that is written in

paragraph form. In this kind of proof, you still need to show

the logical steps of your argument and give the reasons for the

logical steps. Complete the paragraph proof below that proves

that .

Use the _______________________ Similarity Postuate.

First, find MN and PN by using the __________________________.

Then, and because the triangles are

__________________________. Next, find the ratios of the lengths

of the corresponding sides.

So the corresponding sides are __________________________.

Because and are _____________ angles, they are

congruent. So by the ______________________________________,

.�LMN ~ �PQN

�QNP�MNL

LN

QN�

MN

PN�

PN � QNMN � LN

PN � ��MN � ��

PN2 � �MN2 � �

PN2 � � � �MN2 � � � �

PN2 � �2 � �2MN2 � �2 � �2

�LMN ~ �PQN

�LMN ~ �PQN

Lesson 7.4 ■ Indirect Measurement 305

© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Modeling a ParkIndirect Measurement

SCENARIO As part of a science fair project, you are making a

fairly accurate model of a local park that is on the edge of a creek

and some tall oak trees. To make the model, you need the approxi-

mate dimensions of these objects. Because it is not reasonable for

you to directly measure the height of a very tall tree, you must come

up with a different method.

Problem 1 How Tall is That Oak Tree?

Use the steps below to experiment with measuring the height of

a tall object. You will need a tape measure, a marker, and a mirror,

making sure the mirror is absolutely flat.

A. Choose an object that you can easily find the height of, such as a

short tree or a lamp. Use a marker to make a dot near the center

of a mirror. Face the object you would like to measure and place

the mirror between yourself and the object. You, the object, and

the mirror should form a straight line. Look into the mirror and

move directly backward until you can see the top of the object

on the dot, as shown below. Place the marker where you

are standing.

B. Measure the distance between the marker and the dot on the

mirror and measure the distance between the dot on the mirror

and the object. Record your results on the figure above.

C. Measure the height of your eyes and the height of the object

and record your results on the figure above.

D. Show that the triangles in the figure are similar. Show all

your work.

�

7.4

ObjectiveIn this lesson,

you will:

■ Use indirect

measurement to

find heights and

widths of objects.

Key Term■ indirect measurement


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Problem 1 How Tall is That Oak Tree?

E. What do you know about the interior angles of the triangles

whose vertices are located at the mirror? Explain your reasoning.


You will find that this relationship between the interior angles of

the triangles holds for objects of any height.

Investigate Problem 11. You go to the park and use the mirror method to gather enough

information to find the height of one of the trees. The figure

below shows your measurements. Find the height of the tree.

Show all your work and use a complete sentence in your answer.

2. A friend wants to try the mirror method on one of the trees.

Your friend finds that the distance between her and the mirror

is 3 feet and the distance between the mirror and the tree is

18 feet. Your friend’s eye height is 60 inches. Draw a diagram

of this situation. Then find the height of this tree. Show all your

work and use a complete sentence in your answer.

5.5 feet

4 feet 16 feet

�

Take NoteRemember that whenever

you are solving a problem

that involves a kind of

measurement like length

(or weight) you may have to

rewrite some measurements

so that they are using the

same units. For instance,

if a problem involves weight,

all of the weights should be

measured in grams.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 1

3. Your friend notices that the tree is casting a shadow and

suggests that you could also use shadows to find the height

of the tree. She lines herself up with the tree’s shadow so that

the tip of her shadow and the tip of the tree’s shadow meet.

She asks you to measure the distance from the tip of the shadow

to her and then measure the distance from her to the tree.

You then draw a diagram of this situation as shown below.

Find the height of the tree. Show all your work and explain how

you found your answer. Use complete sentences in your answer.

5.5 feet

15 feet 6 feet

�


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Problem 2 How Wide is the Creek?

It is not reasonable for you to directly measure the width of a creek,

but you can measure the width indirectly. You stand on one side of

the creek and your friend stands directly across the creek from you

on the other side as shown in the figure.

A. Your friend is standing 5 feet from the creek and you are standing

5 feet from the creek. Mark these measurements on the diagram

above.

B. You and your friend walk away from each other in opposite

parallel directions. Your friend walks 50 feet and you walk

12 feet. Mark these measurements on the diagram above.

Draw a line segment that connects your starting point and

ending point and draw a line segment that connects your

friend’s starting point and ending point.

C. Draw a line segment that connects you and your friend’s

starting points and draw a line segment that connects you and

your friend’s ending points. Label any angle measures and any

angle relationships that you know on the diagram. Use complete

sentences to explain how you know these angle measures.

D. How do you know that the triangles formed by the lines are

similar? Use a complete sentence to explain your reasoning.

You

Your friend


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 21. Find the distance from your friend’s starting point to your side

of the creek. Show all your work and round your answer to the

nearest tenth, if necessary.

What is the width of the creek? Use complete sentences to

explain how you found your answer.

2. There is also a ravine (a deep hollow in the earth) on another

edge of the park. You and your friend take measurements like

those in Problem 2 to indirectly find the width of the ravine.

The figure below shows your measurements. Find the width of

the ravine. Show all your work and use a complete sentence in

your answer.

You

Your friend

6 feet

60 feet

15 feet

8 feet


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 23. There is also a large pond in the park. A diagram of the pond is

shown below. You want to find the distance across the widest

part of the pond, labeled as . To indirectly find this distance,

you first place a stake at point A. You chose point A so that you

can see the edge of the pond on both sides at points D and E,

where you also place stakes. Then you tie string from point A

to point D and from point A to point E. At a narrow portion of the

pond, you place stakes at points B and C along the string so that

is parallel to . The measurements you make are shown

on the diagram. Find the distance across the widest part of the

pond. Show all your work and use a complete sentence in

your answer.

A

B C

D E

16 feet

20 feet

35 feet

DEBC

DE

Lesson 7.5 ■ Similar Solids 311

© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Making Plastic ContainersSimilar Solids


you will:

■ Find the scale factor of

similar solids.

■ Compare the volumes

and surface areas of

similar solids.

■ Find the dimension of

a similar solid given

the scale factor.

Key Terms■ similar solids

■ scale factor

7.5

SCENARIO One way a plastic container can be made is by

forcing liquid heated plastic into a mold and injecting air into the

mold to form the container. This method is used to make containers

in a variety of shapes, such as cylinders and prisms.

Problem 1 Comparing Containers

Two plastic containers in the shape of rectangular prisms are

shown below.

A. Write and simplify a ratio that compares the length of the base of

container A to the length of the base of container B.

B. Write a ratio that compares the width of the base of container A

to the width of the base of container B.

C. Write a ratio that compares the height of container A to the

height of container B.

D. What do you notice about the ratios? Use a complete sentence

in your answer.

B

6 inches

6 inches

9 inches

A

4 inches4 inches

6 inches


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 11. Just the Math: Similar Solids Containers A and B are

similar solids. Two solids with the same shape are similar if the

ratios of their corresponding measures (length, width, height,

radius) are equal. This ratio is often called the scale factor of

one solid to another solid. What is the scale factor of container

A to container B? Use a complete sentence in your answer.

2. Find the surface areas of container A and container B. Show all

your work and use a complete sentence in your answer.

Find the ratio of the surface area of container A to the surface

area of container B. Simplify the ratio. Show all your work and


3. Find the volumes of container A and container B. Simplify

the ratio. Show all your work and use a complete sentence

in your answer.

Find the ratio of the volume of container A to the volume of

container B. Simplify the ratio. Show all your work and use

a complete sentence in your answer.

4. How do the ratios in Question 2 and Question 3 compare to the

ratio in Question 1? Use complete sentences in your answer.

A

4 inches4 inches

6 inches

B

6 inches

6 inches

9 inches

Take NoteRemember that the volume

of a rectangular prism is

where V is the

volume, B is the area of the

base, and h is the height.

The surface area of a rectan-

gular prism is

where S is the surface area,

B is the area of the base,

P is the perimeter of the

base, and h is the height.

S � 2B � Ph

V � Bh


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 1Why do think that the ratios are related in this way?


5. The two cylindrical plastic containers below are similar.

Complete the following ratios. Write each ratio in simplest form.

Radius of cylinder C : Radius of cylinder D

Height of cylinder C : Height of cylinder D

What is the scale factor? Use a complete sentence in

your answer.

Find the surface areas and volumes of the cylinders.

Leave your answers in terms of .

Complete the ratios. Write each ratio in simplest form.

Surface area of cylinder C : Surface area of cylinder D

Volume of cylinder C : Volume of cylinder D

How does the ratio of the surface areas compare to the scale

factor? Use a complete sentence in your answer.

��

��

�

��

��

D

70 millimeters

60 millimetersC 72 millimeters

84 millimeters

Take NoteRemember that the volume

of a cylinder is

where V is the volume, r is

the radius and h is the

height. The surface area of a

cylinder is

where S is the surface area,

r is the radius, and h is

the height.

S � 2�r 2 � 2�rh

V � �r 2h


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Investigate Problem 1How does the ratio of the volumes compare to the scale factor?


6. For any two similar solids, how does the ratio of the surface

areas compare to the scale factor? Use a complete sentence

in your answer.

For any two similar solids, how does the ratio of the volumes

compare to the scale factor? Use a complete sentence in

your answer.

7. Cube M is similar to cube N with a scale factor of 1 : 4.

The length of the base of cube M is 5 inches. Find the length

of the base of cube N.

What is the ratio of the surface area of cube M to the surface

area of cube N? What is the ratio of the volume of cube M to the

volume of cube N? Use complete sentences to explain how you

found your answers.


© 2

008 C

arn

eg

ie L

earn

ing

, In

c.

7

Date post:	03-Apr-2022
Category:	Documents
Upload:	others
View:	1 times
Download:	0 times

Looking Ahead to Chapter 7 - Michele Clark

Documents