9 4 proportional parts and triangles

Parallel Lines And Proportional Parts & Parts Of Similar Triangles

C hapter 7

A 3

G lencoe G

eom etry

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Answers

Chapter 7 9 Glencoe Geometry

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Practice Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

7-1

1. NUTRITION One ounce of cheddar cheese contains 9 grams of fat. Six of the grams of fat are saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese. 2:3

2. FARMING The ratio of goats to sheep at a university research farm is 4:7. The number of sheep at the farm is 28. What is the number of goats? 16

3. ART Edward Hopper’s oil on canvas painting Nighthawks has a length of 60 inches and a width of 30 inches. A print of the original has a length of 2.5 inches. What is the width of the print? 1.25 in.

Solve each proportion.

4. !58! " !1 x 2! 7.5 5. !1.

x 12! " !

1 5! 0.224 6. !2

6 7 x ! " !

4 3! 6

7. !x #3 2

! " ! 8 9! !

2 3! 8. !

3x 4 $ 5 ! " !

$ 7 5 ! !

5 7! 9. !

x $ 4

2 ! " !

x # 2

4 ! "10

Find the measures of the sides of each triangle.

10. The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet. 24 ft, 32 ft, 48 ft

11. The ratio of the measures of the sides of a triangle is 7:9:12, and its perimeter is 84 inches. 21 in., 27 in., 36 in.

12. The ratio of the measures of the sides of a triangle is 6:7:9, and its perimeter is 77 centimeters. 21 cm, 24.5 cm, 31.5 cm

Find the measures of the angles in each triangle.

13. The ratio of the measures of the angles is 4:5:6. 48, 60, 72

14. The ratio of the measures of the angles is 5:7:8. 45, 63, 72

15. BRIDGES The span of the Benjamin Franklin suspension bridge in Philadelphia, Pennsylvania, is 1750 feet. A model of the bridge has a span of 42 inches. What is the ratio of the span of the model to the span of the actual Benjamin Franklin Bridge? !5

1 00!

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Chapter 7 8 Glencoe Geometry

Skills Practice Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

7-1

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1. FOOTBALL A tight end scored 6 touchdowns in 14 games. Find the ratio of touchdowns per game.

0.43:1

2. EDUCATION In a schedule of 6 classes, Marta has 2 elective classes. What is the ratio of elective to non-elective classes in Marta’s schedule?

1:2

3. BIOLOGY Out of 274 listed species of birds in the United States, 78 species made the endangered list. Find the ratio of endangered species of birds to listed species in the United States.

!1 3 3 9 7!

4. ART An artist in Portland, Oregon, makes bronze sculptures of dogs. The ratio of the height of a sculpture to the actual height of the dog is 2:3. If the height of the sculpture is 14 inches, find the height of the dog.

21 in.

5. SCHOOL The ratio of male students to female students in the drama club at Campbell High School is 3:4. If the number of male students in the club is 18, what is the number of female students?

24

Solve each proportion.

6. !25! " !4 x 0! 16 7. !1

7 0! " !

2 x 1 ! 30 8. !25

0 ! " !

4 6 x ! 6

9. !54 x ! " !

3 8 5 ! 3.5 10. !x #3

1 ! " !

7 2! 9.5 11. !

1 3 5 ! " !

x $ 5

3 ! 28

Find the measures of the sides of each triangle.

12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters. 90 cm, 150 cm, 210 cm

13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters. 55 m, 66 m, 99 m

14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet. 28 ft, 42 ft, 56 ft

15. The ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches. 10 in., 14 in., 16 in.

Answ ers (L

esson 7–1)

C hapter 7

A 7

G lencoe G

eom etry

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Answers

Chapter 7 17 Glencoe Geometry

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Practice Similar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

7-2

Determine whether each pair of figures is similar. Justify your answer.

1. 2.

JKLM ! QRSP ; !ABC ! !UVT; "A " "U, "J " "Q, "K " "R, "B " "V, and "C " "T by the "L " "S, "M " "P, and Third Angle Theorem and !Q JK

R! " !R K

S L ! " !

L S M P! " !P

M Q J ! " !

5 3! or 1.67 !

A U

B V! " !

B V

C T! " !

C TU

A ! " !

2 3!.

Each pair of polygons is similar. Write a similarity statement, and find x, the measure(s) of the indicated side(s), and the scale factor.

3. L!M! and M!N! 4. D!E! and D!F!

ABCD ! LMNP ; !ABC ! !DEF; ! 3 2!; 10 !

1 2!; 7!

1 2!; !

4 3! 7; 4; 8; !

3 2!

5. COORDINATE GEOMETRY Triangle ABC has vertices A(0, 0), B(!4, 0), and C(!2, 4). The coordinates of each vertex are multiplied by 3 to create !AEF. Show that !AEF is similar to !ABC.

AB " 4, BC " CA " 2#5$ and AE " 12, EF " FA " 6#5$. ! A A

B E! " !

B E

C F! " !

C FA

A ! " !

1 3!; "A " "A and since B$C$ || E$F$, "B " "E and

"C " "F. Since the corresponding angles are congruent and the corresponding sides are proportional, !ABC ! !AEF.

6. INTERIOR DESIGN Graham used the scale drawing of his living room to decide where to place furniture. Find the dimensions of the living room if the scale in the drawing is 1 inch " 4.5 feet. 18 ft by 11 ft 3 in. 4 in.

21–2 in.

6 12

40#

40#

x $ 1

x % 3A F

E

D

B C14

10

x $ 6

x $ 9

C N MD

A B LP

2412

14 16 2118

VUAC

B T

25 12

9

14.4

15 24

20

15

JM Q R

S P

KL

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Chapter 7 16 Glencoe Geometry

Skills Practice Similar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

7-2

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Determine whether each pair of figures is similar. Justify your answer.

1. 2.

!ABC ! !DEF; "A " "D, rhombus PQRS ! rhombus WXYZ; "C " "F, and "B " "E "P " "W, "Q " "X, "R " "Y, because if two angles of one "S " "Z; triangle are congruent to two

!W PQ

X! " ! Q XY

R ! " !

R Y

S Z! " !Z

S W P ! " !7

3 .5! " 0.4angles of a second triangle, then

the third angles are congruent;

! A D

B E! " !

B E

C F! " !

C FD

A ! " !

3 2!.

Each pair of polygons is similar. Write a similarity statement, and find x, the measure(s) of the indicated side(s), and the scale factor.

3. G!H! 4. S!T! and S!U!

ABCD ! EFGH; !WXY !!STU (or !SUT ) 6!12!; 6!

1 2!; 2 7; 12; 12; !

1 3!

5. W!T! 6. T!S! and S!P!

PQRS ! UVWT; !LMN! !TSP; 5; 6; !32! 13; 12; 15; !

2 3!

10

8

x $ 2

x % 1

SP

M N

L

T

9

10 3

x $ 1

5

P

S

R U V

WT

Q

x $ 5

x $ 5

9

3

44 S

W

XY

T

U26

14

12 13

B C

DA

13

7 6 x

G

E H

F

7.5

7.5

7.5 7.5

Z Y

XW

3

3

33 S R

QP

10.5

6 9

59# 35#A C

B

7

4 6 59# 35#D F

E

Answ ers (L

esson 7–2)

C hapter 7

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G lencoe G

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Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 7 23 Glencoe Geometry

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Skills Practice Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-3

Determine whether each pair of triangles is similar. Justify your answer.

1. 2.

No; the sides are not proportional. yes; !ABC ! !PQR (or !QPR ); SSS Similarity

3. 4.

yes; !STU ! !JPM; yes; !SKM ! !RTQ; SAS Similarity AA Similarity

ALGEBRA Identify the similar triangles, and find x and the measures of the indicated sides.

5. A!C! and E!D! 6. J!L! and L!M!

!ABC ! !DBE; 15; 16; 20 !JKL ! !MNL; 10; 28; 7

7. E!H! and E!F! 8. V!T! and S!T!

!DEF ! !GEH; 4; 9; 18 !RST ! !UVT; 7.5; 1.5; 5

6

14

S

U

V

R T

x % 4

x % 6 12

9

6

9

D

E

F H

Gx $ 5

16

4 M

NL

J

K

x $ 18

x % 3

15 12

D

E CB

A x $ 1

x $ 5

30#60# RQ

M

S

K

T

21 70#

15

MJ

P

1470#10

U

S

T

12

128

9

9 6

C P

QR

A

B

2014

13 12 21

9

S W X

Y

R

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Chapter 7 22 Glencoe Geometry

Study Guide and Intervention (continued) Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-3

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Use Similar Triangles Similar triangles can be used to find measurements.

!ABC ! !DEF. Find x and y.

#D AC

F# " # B E

C F# #D

AB E# " #

B E

C F#

" # 1 9 8 # #1

y 8# " #

1 9 8 #

18x " 9(18"3!) 9y " 324 x " 9"3! y " 36

18"3! #x

18"#3

B

CA

18 18 9 y

x

E

FD

A person 6 feet tall casts a 1.5-foot-long shadow at the same time that a flagpole casts a 7-foot-long shadow. How tall is the flagpole?

The sun’s rays form similar triangles. Using x for the height of the pole, #6x# " #

1 7 .5 #,

so 1.5x " 42 and x " 28. The flagpole is 28 feet tall.

7 ft

6 ft ?

1.5 ft

Each pair of triangles is similar. Find x and y.

1. 2.

x " 26; y " 17.5 x " 24; y " 30

3. 4.

x " 12; y " 24#2$ x " 19.3; y " 46

5. 6.

x " 18; y " 10.8 x " 10!23!; y " 16! 1 2!

7. The heights of two vertical posts are 2 meters and 0.45 meter. When the shorter post casts a shadow that is 0.85 meter long, what is the length of the longer post’s shadow to the nearest hundredth? 3.78 m

y

x32

10

2230y x

9

8

7.2

16

yx

60 38.6

23

30 24

36"#2 yx

3636

20

13 26

y

x $ 2 35

13

10 20 y

x

Example 1 Example 2

Exercises

Answ ers (L

esson 7–3)

C hapter 7

A 11

G lencoe G

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Answers

Chapter 7 25 Glencoe Geometry

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Word Problem Practice Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-3

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1. MINIATURES Marla likes her chair so much that she decides to make a miniature replica of it for her pet hamster. Find the value of x.

3.75 in.

2. MODELS Jim has a scale model of his sailboat. The figure shows drawings of the original sailboat and the model. Find x.

160 in.

3. GEOMETRY Georgia draws a regular pentagon and starts connecting its vertices to make a 5-pointed star. After drawing three of the lines in the star, she becomes curious about two triangles that appear in the figure, !ABC and !BCE. They look similar to her. Prove that this is the case. Sample answer: m!ADB ! 108, so m!DBA ! 36 (base of iso- sceles "ABD). Thus, m!ABC ! 72°. Similarly, m!DCB ! 36° and m!ACB ! 72°. So, m!BEC ! 72° and m!BAC ! 36°. Therefore, "ABC and "BCE are similar.

4. SHADOWS A radio tower casts a shadow 8 feet long at the same time that a vertical yardstick casts a shadow half an inch long. How tall is the radio tower? 576 ft

MOUNTAIN PEAKS For Exercises 5 and 6, use the following information. Gavin and Brianna want to know how far a mountain peak is from their houses. They measure the angles between the line of site to the peak and to each other’s houses and carefully make the drawing shown.

The actual distance between Gavin and Brianna’s house is 1!12! miles.

5. What is the actual distance of the mountain peak from Gavin’s house? Round your answer to the nearest tenth of a mile. 12.2 mi

6. What is the actual distance of the mountain peak from Brianna’s house? Round your answer to the nearest tenth of a mile. 12.3 mi

Gavin

Brianna

Peak

2 in.

2.015 in. 0.246 in.

90˚ 83˚

A

B

D

E

C

52˚

203 in.

10.15 in. 8 in. 52˚

x in.

14”

3.5”

15”

x

Chapter 7 24 Glencoe Geometry

Practice Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-3

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Determine whether each pair of triangles is similar. Justify your answer.

1. 2.

yes; "JAK ! "WSY; No; the sides are not proportional. SAS Similarity

ALGEBRA Identify the similar triangles, and find x and the measures of the indicated sides.

3. L!M! and Q!P! 4. N!L! and M!L!

"LMN ! "PQN; 9; 12; 8 "JLN ! "KLM; 2; 21; 14

Use the given information to find each measure.

5. If T!S! || Q!R!, TS " 6, PS " x # 7, 6. If E!F! || H!I!, EF " 3, EG " x # 1, QR " 8, and SR " x $ 1, HI " 4, and HG " x # 3, find PS and PR. find EG and HG.

PS ! 12; PR ! 16 EG ! 6; HG ! 8

INDIRECT MEASUREMENT For Exercises 7 and 8, use the following information. A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow.

7. Write a proportion that can be used to determine the height of the lighthouse.

Sample answer: If x ! height of lighthouse, "5. x 25" ! "

12 8 8

".

8. What is the height of the lighthouse? 84 ft

F

G E

H

I

S RP

Q T

8

N

J LK

M x # 5

6x # 2

24

12

18 N

P

QL

M

x # 3 x $ 1

18

16 12.5 16

14

11

M

L N

S R

T 42%

42%

24

18 12

16J

A K

Y S

W

Answ ers (L

esson 7–3)

C hapter 7

A 14

G lencoe G

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Copyright ©Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter 7 31 Glencoe Geometry

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Practice Parallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

7-4

1. If AD " 24, DB " 27, and EB " 18, 2. Find x, QT, and TR if QT " x # 6, find CE. SR " 12, PS " 27, and TR " x $ 4.

16 12; 18; 8

Determine whether J"K" || N"M".

3. JN " 18, JL " 30, KM " 21, and ML " 35 no

4. KM " 24, KL " 44, and NL " !56!JN yes

COORDINATE GEOMETRY For Exercises 5 and 6, use the following information. Triangle EFG has vertices E($4, $1), F(2, 5), and G(2, $1). Point K is the midpoint of E!G! and H is the midpoint of F!G!.

5. Show that E!F! is parallel to K!H!. The slope of E"F" is 1 and the slope of K"H" is 1, so E"F" is parallel to K"H".

6. Show that KH " !12!EF.

The length of K"H" is 3#2", and the length of E"F" is 6#2".

7. Find x and y. 8. Find x and y.

x ! 4, y ! 9 x ! "52", y ! " 9 4"

9. MAPS The distance from Wilmington to Ash Grove along Kendall is 820 feet and along Magnolia, 660 feet. If the distance between Beech and Ash Grove along Magnolia is 280 feet, what is the distance between the two streets along Kendall? about 348 ft

Ash Grove

Beech

Magnolia Kendall

Wilmington

4– 3y # 2

3x $ 4 x # 1

4y $ 4 3x $ 4 2–3y # 3

1– 3y # 6

5– 4x # 3

x

y

O

E G

F

H

K

N LJ

K M

S

Q

T RP

D

E

C

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Chapter 7 30 Glencoe Geometry

Skills Practice Parallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

7-4

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1. If JK " 7, KH " 21, and JL " 6, 2. Find x and TV if RU " 8, US " 14, find LI. TV " x $ 1 and VS " 17.5.

18 11; 10

Determine whether B!C! || D!E!. 3. AD " 15, DB " 12, AE " 10, and EC " 8

yes

4. BD " 9, BA " 27, and CE is one third of EA no

5. AE " 30, AC " 45, and AD is twice DB yes

COORDINATE GEOMETRY For Exercises 6–8, use the following information. Triangle ABC has vertices A($5, 2), B(1, 8), and C(4, 2). Point D is the midpoint of A!B! and E is the midpoint of A!C!.

6. Identify the coordinates of D and E. D($2, 5); E($0.5, 2)

7. Show that B!C! is parallel to D!E!. Since the slope of B"C" is $2 and the slope of D"E" is $2, B"C" is parallel to D"E".

8. Show that DE " !12!BC.

The length of D"E" is #11.25" or or !3# 2

5"! and the length of B"C" is #45" or 3#5".

9. Find x and y. 10. Find x and y.

x ! 6, y ! 6.5 x ! 2, y ! 4

3– 2x # 2

x # 3 2y $ 1

3y $ 5 2x # 1 3y $ 8

y # 5x # 7

#45"" 2

x

y

O

A C

B

D

E

C

B

E

D

A

R

U V

T

S

H L

K J

I

Answ ers (L

esson 7–4)

C hapter 7

A 17

G lencoe G

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Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Answers

Chapter 7 37 Glencoe Geometry

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Skills Practice Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-5

Find the perimeter of the given triangle.

1. !JKL, if !JKL ! !RST, RS ! 14, 2. !DEF, if !DEF ! !ABC, AB ! 27, ST ! 12, TR ! 10, and LJ ! 14 BC ! 16, CA ! 25, and FD ! 15

50.4 40.8

3. !PQR, if !PQR ! !LMN, LM ! 16, 4. !KLM, if !KLM ! !FGH, FG ! 30, MN ! 14, NL ! 27, and RP ! 18 GH ! 38, HF ! 38, and KL ! 24

38 84.8

Use the given information to find each measure.

5. Find FG if !RST ! !EFG, S"H" is an 6. Find MN if !ABC ! !MNP, A"D" is an altitude of !RST, F"J" is an altitude of altitude of !ABC, M"Q" is an altitude of !EFG, ST ! 6, SH ! 5, and FJ ! 7. !MNP, AB ! 24, AD ! 14, and MQ ! 10.5.

8.4 18

Find x.

7. !HKL ! !XYZ 8.

8!13! 8.4

LH

K

18

10

Y

15

x

ZX 7

20

24

x

C

A

BD P NQ

M

J GE

F

H TR

S

G L

K

M

F

H

L M P Q

R N

A

B

C

D

E

F

R S

T

J K

L

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Chapter 7 36 Glencoe Geometry

Study Guide and Intervention (continued) Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-5

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Special Segments of Similar Triangles When two triangles are similar, corresponding altitudes, angle bisectors, and medians are proportional to the corresponding sides. Also, in any triangle an angle bisector separates the opposite side into segments that have the same ratio as the other two sides of the triangle.

In the figure, !ABC ! !XYZ, with angle bisectors as shown. Find x.

Since !ABC ! !XYZ, the measures of the angle bisectors are proportional to the measures of a pair of corresponding sides.

" A X

B Y" ! "Y

B W D "

" 2 x 4 " ! "

1 8 0 "

10x ! 24(8) 10x ! 192

x ! 19.2

24 10

D W

Y

ZX C

B

A

8 x

S"U" bisects "RST. Find x.

Since S"U" is an angle bisector, "RTU U " ! "

R TS

S ".

"2 x 0" ! "

1 3 5 0"

30x ! 20(15) 30x ! 300

x ! 10

x 20

3015

U T

S

R

Find x for each pair of similar triangles.

1. 2.

10 8

3. 4.

2.25 8.75

5. 6.

12!56! 15

2x " 2 x # 1

2816 17

11

x # 7

x

10

10x

8

873

3 4

x

69

12 x

x

36

18 20

Example 1 Example 2

Exercises

Answ ers (L

esson 7–5)

C hapter 7

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Chapter 7 39 Glencoe Geometry

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Word Problem Practice Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-5

Le ss

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1. JOGGING Macy wants to jog around Triangle Park. On a scale map, the park has a perimeter of 11 inches. The scale of the map is 1 inch : 100 yards. What is the perimeter of the actual park? 1100 yd

2. TENTS Jana went camping and stayed in a tent shaped like a triangle. In a photo of the tent, the base of the tent is 6 inches and the altitude is 5 inches. The actual base was 12 feet long. What was the height of the actual tent?

10 ft

3. PLAYGROUND The playground at Hank’s school has a large right triangle painted in the ground. Hank starts at the right angle corner and walks toward the opposite side along an angle bisector and stops when he gets to the hypotenuse.

How much farther from Hank is point B versus point A?

7!17! ft

4. FLAG POLES A flag pole attached to the side of a building is supported with a network of strings as shown in the figure.

The rigging is done so that AE ! EF, AC ! CD, and AB ! BC. What is the ratio of CF to BE? 2 : 1

COPIES For Exercises 5 and 6, use the following information. Gordon made a photocopy of a page from his geometry book to enlarge one of the figures. The actual figure that he copied is shown below.

The photocopy came out poorly. Gordon could not read the numbers on the photocopy, although the triangle itself was clear. Gordon measured the base of the enlarged triangle and found it to be 200 millimeters.

5. What is the length of the drawn altitude of the enlarged triangle? Round your answer to the nearest millimeter. 149 mm

6. What is the length of the drawn median of the enlarged triangle? Round your answer to the nearest millimeter. 154 mm

39 mm

M ed

ian

Al tit

ud e

30 mm

29 mm

A B C

D F

E

A

B

45˚

30 ft

40 ft

50 ft

x

Chapter 7 38 Glencoe Geometry

Practice Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

7-5

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cG raw

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Find the perimeter of the given triangle.

1. !DEF, if !ABC ! !DEF, AB ! 36, 2. !STU, if !STU ! !KLM, KL ! 12, BC ! 20, CA ! 40, and DE ! 35 LM ! 31, MK ! 32, and US ! 28

93.3" 65.625

Use the given information to find each measure.

3. Find PR if !JKL ! !NPR, K"M" is an 4. Find ZY if !STU ! !XYZ, U"A" is an altitude of !JKL, P"T" is an altitude of altitude of !STU, Z"B" is an altitude of !NPR, KL ! 28, KM ! 18, and !XYZ, UT ! 8.5, UA ! 6, and PT ! 15.75. ZB ! 11.4.

24.5 16.15

Find x.

5. 6.

13.5 16.8

PHOTOGRAPHY For Exercises 7 and 8, use the following information. Francine has a camera in which the distance from the lens to the film is 24 millimeters.

7. If Francine takes a full-length photograph of her friend from a distance of 3 meters and the height of her friend is 140 centimeters, what will be the height of the image on the film? (Hint: Convert to the same unit of measure.) 11.2 mm

8. Suppose the height of the image on the film of her friend is 15 millimeters. If Francine took a full-length shot, what was the distance between the camera and her friend? 2.24 m

x 28

20 30402x # 1 25x # 4

YX B

Z

TS A

U

MJ L

K

TN R

P

K S

L T

UMB E

CFDA

Answ ers (L

esson 7–5)