Lesson 6.5 Similarity and Measurement 293

LE

SS

ON

6.5Review Vocabularyratio, p. 269proportion, p. 275similar figures, p. 288

B E F O R E Now W H Y ?

You identified similarfigures.

You’ll find unknown sidelengths of similar figures.

So you can find the height of apalm tree, as in Ex. 5.

Similarity and Measurement

Cactus A man who is 6 feet tall is standing near a saguaro cactus. The length of the man’s shadow is 2 feet. The cactus casts a shadow 5 feet long. How tall is the cactus?

In Example 2, you will see how to use similar triangles to measure the cactus’s height indirectly.

Given ABCDS EFGH, find EH.

Solution

Use the ratios of the lengths of correspondingsides to write a proportion involving theunknown length, EH.

�BFG

C� � �

AEH

D� Write proportion involving EH.

�13

20� � �

1x6� Substitute.

12x � 30 p 16 Cross products property

12x � 480 Multiply.

x � 40 Divide each side by 12.

Answer The length of EH&* is 40 inches.

Example 1 Finding an Unknown Side Length in Similar Figures

Checkpoint

1. Given TSTUSTDEF, find DF. 2. Given JKLMS PQRS, find PQ.

R Q

M L

21 in.

x

S P35 in.J K36 in.T

S U35 mm

45 mm

E

D Fx

36 mm

G

H

F

30 in.

E

x

16 in.A D

12 in.B C

294 Chapter 6 Ratio, Proportion, and Probability

Indirect Measurement You can use similar figures to find lengths that aredifficult to measure directly.

Given TABC S TDEC, find BE.

To find BE, write and solve a proportion.

�DAB

E� � �

BEC

C� Write proportion.

�DAB

E� � �

BEE�C

EC� Use the fact that BC � BE � EC.

�21

05� � �

x �36

36� Substitute.

20 p 36 � 15(x � 36) Cross products property

720 � 15x � 540 Multiply.

180 � 15x Subtract 540 from each side.

12 � x Divide each side by 15.

Answer The length of BE&* is 12 inches.

Example 3 Using Algebra and Similar Triangles

Use indirect measurement to find theheight of the cactus described on page 293. The cactus and the man areperpendicular to the ground. The sun’srays strike the cactus and the man at thesame angle, forming two similar triangles.

Solution

Write and solve a proportion to find the height h (in feet) of the cactus.

�HHeeigighhttooffcmac

atun

s��

�h6

� � �52

� Substitute.

2h � 6 p 5 Cross products property

2h � 30 Multiply.

h � 15 Divide each side by 2.

Answer The cactus has a height of 15 feet.

Length of cactus’s shadow���

Length of man’s shadow

Example 2 Using Indirect Measurement

Checkpoint

3. A cactus is 5 feet tall and casts a shadow that is 1.5 feet long. How tallis a nearby cactus that casts a shadow that is 8 feet long?

36 in.B CEx

AD

20 in.15 in.

h

5 ft

6 ft

2 ft

In the Real WorldCactus To live in the desert, acactus contains an enormousquantity of water inside itstrunk. Suppose a saguarocactus weighs 8000 poundsand holds 1 ton of water. Whatfraction of the cactus’s weight iswater?

To solve a problem like the onein Example 3, you may find ithelpful to redraw the twotriangles so they don’t overlap.

Study Strategy

6. Given TABCSTLMN, find LN. 7. Given LMNPS QRST, find RS.

8. Given ABCDS KLMJ, find KL. 9. Given ABCDS RSTU, find UR.

B A

15 yd25 yd

x

C D T S

30 yd

U RD

AC

B

11.2 cm

J

KM

14 cm

7.2 cm

Lx

45 mm

N

P

M

L

25 mm

27 mm

S

T

R

Q

x

24 in. 16 in.

L N

M

x

A C

B

36 in.

Practice and Problem Solving

1. EFGH S JKLM. Which side of EFGH corresponds with JM&* ?2. Describe how similar triangles are useful for indirect measurement.

In Exercises 3 and 4, TABC S TDEF.

3. Find EF.

4. Find FD.

5. Palm Tree A man who is 74 inches tall stands beside apalm tree. The length of theman’s shadow is 26 inches. Thepalm tree’s shadow is 80 incheslong. How tall is the palm tree?

The rays of the sun createsimilar triangles for theman and the palm tree.Write a proportion usingthe triangles.

Find the height of the palm tree to the nearest inch.2

1

Guided Problem Solving

Skill Check

Vocabulary Check

Guided Practice

ExercisesMore Practice, p. 808

6.5 eWorkbook PlusCLASSZONE.COM

INTERNET

17 mA

E

DC

B F16 m

8 m 8 m

Lesson 6.5 Similarity and Measurement 295

x

80 in.

74 in.

26 in.

Example Exercises1 6–9, 132 12, 143 10, 11, 15

Homework Help

Online ResourcesCLASSZONE.COM

• More Examples• eTutorial Plus

296 Chapter 6 Ratio, Proportion, and Probability

Find the length of DE&*.

10. TABCSTADE 11. ABCDS AGFE

12. Bryce Canyon Bryce Canyon National Park in Utah is known for itsunusual rock formations. One rock casts a shadow 21 feet long. A girlstanding near this rock is 5 feet 3 inches tall and casts a shadow 7 feet long.

a. Convert Write the girl’s height in inches.

b. Write and solve a proportion to find the height of the rock in feetand inches.

13. Poster You are enlarging a photograph to make a poster. The posterwill be similar to the original photograph. The photograph is 6 inches tall and 4 inches wide. The poster will be 2.5 feet wide. How tall will the poster be? Find the poster’s perimeter.

14. Surveying You can use indirect measurement to find the distanceacross a river by following these steps.

Start at a point A directly across the river from a landmark, such as a tree, at point B.

Walk 30 feet along the river topoint C and place a stake.

Walk 20 feet farther along theriver to point D.

Turn and walk directly awayfrom the river. Stop at point E,where the stake you plantedlines up with the landmark.TABC STDEC.

Suppose you walk 24 feet away from the river along DE&* before thestake lines up with the landmark. Write and solve a proportion tofind the distance AB across the river.

4

3

2

1

18 m xA D

CB

G

E

7 m

14 m

F

9 ft 6 ftA C

B

E

D10 ft

x

Hikers in Bryce CanyonNational Park, Utah

h

21 f t7 f t

5 f t 3 in.

xA

BC

D

EF

A

B

CD

E

30 f t20 f t

Tree

Stake

24 f t

You may find it helpful to drawa diagram when solvingproblems about similar figures.For help with drawing adiagram, see p. 795.

Review Help

Lesson 6.5 Similarity and Measurement 297

Arrange the four congruent triangles to create a larger similar triangle. Arrange the four congruent rectangles to create a larger similar rectangle. Sketch your answers.

Can you arrange the four congruent quadrilaterals in the bottom row tocreate a larger similar quadrilateral?

Putting the Pieces Together

15. In the figure, TABC, TADE, and TAFG are all similar.

a. Find DE and FG.

b. Find AE and AG.

16. Critical Thinking Given TABCSTDEF, tell whether the giveninformation is enough to find the specified measurements. Explainyour thinking.

a. You know AB, BC, CA, and DE. You want to find EF and FD.

b. You know AB, BC, and FD. You want to find CA.

c. You know maB. You want to find maE.

17. Challenge A rectangular box has a length of 3 inches, a width of 2 inches, and a height of 4 inches. Find the dimensions of three similarboxes: one that has a length of 6 inches, one that has a width of 6 inches, and one that has a height of 6 inches.

Solve the equation. (Lesson 5.6)

18. �57

�x � 16 19. ��49

�x � 12 20. �34

�x � �58

� � 1 21. ��12

�x � 6 � 5 �45

�

Order the ratios from least to greatest. (Lesson 6.1)

22. 5 : 3, 6 : 4, �131�, 3 to 1, 33 : 100 23. 15 to 9, 35 : 25, �

43

43�, �

22

20�, 8 : 3

In Exercises 24 and 25, TRST S TVUT.

24. Multiple Choice Find UV.

A. 3 yards B. 4 yards

C. 4.5 yards D. 5 yards

25. Multiple Choice Find ST.

F. 8 yards G. 10 yards H. 12 yards I. 15 yards

Standardized TestPractice

Mixed Review

12 cm

6 cm

F ABD

G E

C5 cm 13 cm

12 cm

8 ydR TV

S

U9 yd

5 yd

4 yd