?

Math On the Spot

my.hrw.com

B

A C12 cm

16 cm

8 cm

103°

48° 29°

K

C x

y

L J56 cm

28 cm

12 cm

cm

103°

29°

ESSENTIAL QUESTION

Finding Unknown Measures in Similar TrianglesWhen you measure the height of a door with a measuring tape, you are using

direct measurement. The process of using similar shapes and proportions to find

a measure is called indirect measurement. You can use indirect measurement

to measure things that are difficult to measure directly, like the height of a tree.

▵ABC ~ ▵JKL. Find the unknown measures.

Find the unknown side, x.

AB __

JK = BC

__ KL

8 __

28 = 12

__ x

8 ÷ 4 _____

28 ÷ 4 = 2 _

7

2 _

7 = 12

__ x

x = 42 cm

Find y.

∠K corresponds to ∠B.

y = 103°

Reflect1. Analyze Relationships What other proportion could be used to find

the value of x in the example? Explain.

EXAMPLEXAMPLE 1

A

B

How can you use similar shapes to find unknown measures?

L E S S O N

4.2 Using Similar Shapes

×6

×6

Proportionality—7.5.A Generalize the critical attributes of similarity, including ratios within and between similar shapes.

7.5.A

Write a proportion using corresponding sides. Side AB corresponds to side JK, and side BC corresponds to side KL.Substitute the known lengths of the sides.

Simplify 8 ___ 28 to help you find a factor of 12. 2 is a factor of 12.Since 2 times 6 is 12, multiply 7 times 6 to find the value of x.

Corresponding angles of similar triangles have equal angle measures.

121Lesson 4.2

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

• Im

age C

redi

ts: ©

Dan

iel g

rill/A

lamy

Math On the Spotmy.hrw.com

B A

J G

H

C

yx

10 cm

6 cm 11.6 cm

5.8 cm

5 cm

59°

31°

J G

H

yx

11.6 cm

5.8 cm

5 cm

s120 cm160 cm

60 cmB A

HC

t

90 cm

m

19°

120°

41°

JG

H

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

50 m

?

18 m

9 m

Determining Unknown Measures in Similar Four-Sided ShapesYou can use similar shapes and proportions to find measures of sides

in rectangles and other four-sided shapes.

A volleyball court is a rectangle that is similar in shape to an Olympic-sized

pool. Find the width of the pool.

Let w = the width of the pool.

18 __

50 = 9 __ w

18 __

50 = 9 __ w

w = 25

The pool is 25 meters wide.

EXAMPLE 2

▵ABC ~ ▵JGH. Find the unknown measures.

YOUR TURN

2.

3.

y = x =

s = t =

÷2

÷2

Math TalkMathematical Processes

7.5.A

With similar triangles, you need to find missing sides

and angles. Why do you need to find only missing

sides with rectangles?

Write a proportion using corresponding sides.

Since18 divided by 2 is 9, divide 50 by 2 to find the value of w.

Unit 2122

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

Math Trainer

Online Assessment and Intervention

Personal

my.hrw.com

54 ft

42 ft ?

36 ft

h

6 ft

1.5 ft4.5 ft

Math On the Spot

my.hrw.com

16 ft

5 ft

4 ft

h

Find the height of the totem pole.

height of person

______________ shadow of person

= height of totem pole

_________________ shadow of totem pole

5 _ 4

= h __ 16

5 _ 4

= h __ 16

h = 20

The totem pole is 20 feet tall.

5. A student who is 4 feet tall stands beside a tree. The tree has a shadow

that is 12 feet long at the same time that the shadow of the student is

6 feet long. Find the height of the tree.

6. A photographer is taking a photo of a statue of

Paul Bunyan, the legendary giant lumberjack.

He measures the length of his shadow and the

shadow cast by the statue. Find the height of the

Paul Bunyan statue.

EXAMPLEXAMPLE 3

YOUR TURN

4. These rectangular

gardens are similar in

shape. Find the width

of the smaller garden.

YOUR TURN

Using Indirect MeasurementWhen you write a proportion to calculate a measurement, you can write ratios

comparing measures of the same object. For example, use the ratio formed by

the height of an object and the length of its shadow.

×4

×4

7.5.A

Write a proportion.

Substitute known values.

Since 4 × 4 = 16, multiply 5 by 4 to find the value of h.

123Lesson 4.2

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

B

A C

z

d

8 m 10 m

6 m

Y

X Z

25 m

15 m

37°

53°

B

C

A Z

Y

X

st

8 in.

12 in.

14 in. 42 in.

36 in.35°

85° 60°

B

CA

Z

Y

X x

t

8 in.

28 in.

7 in.

11 in.

44 in.

92°

42° 46°

A

BC

ZY

X

x

a18 ft 18 ft72 ft 72 ft

25 ft

48°

84°

48°

y

x9.5 in. 19 in.

19 in.

19 in. 19 in.

90° 90°

90° 90°

12 cm

6 cm

Q R

T Sf

q

5 cm

A B

D C

Guided Practice

The rectangles in each pair are similar. Find the unknown measures. (Example 2)

5. 6.

▵ABC ~ ▵XYZ in each pair. Find the unknown measures (Example 1)

1. 2.

3. 4.

7. Samantha wants to find the height of a pine tree in her yard. She

measures the height of the mailbox at 3 feet and its shadow at 4.8 feet.

Then she measures the shadow of the tree at 56 feet. How tall is the tree?

(Example 3)

8. Describe how you could use your height and a yardstick to determine the

unknown height of a flagpole.

ESSENTIAL QUESTION CHECK-IN??

Unit 2124

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

J

N Z

A

K O

RY

X

ZP Q

ab

20 cm9 cm

8 cm

12 cm

30 cm

40°

89°

R

Y Q

X Z

P

s

y

48 m 64 m40 m30 m

35 m

58°

77°

45°

15 ft 5 ft

3 ft

x

Personal Math Trainer

Online Assessment and

Interventionmy.hrw.com

Name Class Date

Independent Practice4.2

9. A cactus casts a shadow that is 15 ft long.

A gate nearby casts a shadow that is 5 ft

long. Find the height of the cactus.

10. Two ramps modeled with triangles

are similar. Which side of triangle KOA

corresponds to _

JN ? Explain.

11. A building with a height of 14 m casts a

shadow that is 16 m long while a taller

building casts a 24 m long shadow. What is

the height of the taller building?

12. Katie uses a copy machine to enlarge her

rectangular design that is 6 in. wide and

8 in. long. The new width is 10 in. What is

the new length?

13. Art An art exhibit at a local museum

features several similarly shaped metal cubes

welded together to make a sculpture. The

smallest cube has a edge length of 6 inches.

a. What are the edge lengths of the other

cubes if the ratios of similarity to the

smallest cube are 1.25, 4 _ 3 , 1.5, 7 _

4 , and 2

respectively?

b. If the artist wanted to add a smaller

cube with an edge length with a ratio

of 2 _ 3 to the sculpture, what size would

the cube be?

c. Why do you only have to find the

length of one edge for each cube?

▵XYZ ~ ▵PQR in each pair. Find the unknown

measures.

14.

15.

7.5.A

125Lesson 4.2

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany

Work Area

16. Two common envelope sizes are 3 1 _ 2 in. × 6 1 _

2 in. and 4 in. × 9 1 _

2 in. Are these

envelopes similar? Explain.

17. A pair of rectangular baking pans come in a set together for $15. One pan

is 13 inches by 9 inches and the other pan is 6 inches by 6 inches. Without

doing any calculations, how can you tell that these pans are not similar?

18. Draw Conclusions In the similar triangles used in indirect measurement

with the shadows of a flagpole and a person, which sides of the triangles

represent the rays of the sun?

19. Make a Conjecture Do you think it is possible to use indirect

measurement with shadows if the sun is directly overhead? Explain.

20. Analyze Relationships Joseph's parents have planted two gardens. One

is square and has an area of 25 ft 2 . The other one has two sides equal to 2 _ 3

of one side of the square, and the other two sides equal to 5 _ 2 of one side

of the square.

a. Find the dimensions of the other garden. Explain how you found

your answer.

b. Find the area of the other garden.

FOCUS ON HIGHER ORDER THINKING

Unit 2126

© H

ough

ton

Miff

lin H

arco

urt P

ublis

hing

Com

pany