chapter 7 resource masters - · pdf filethe chapter 7 resource masters includes the core...

Chapter 7 Resource Masters

Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.

ISBN10 ISBN13

Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4

Homework Practice Workbook 0-07-890849-3 978-0-07-890849-1

Spanish Version

Homework Practice Workbook 0-07-890853-1 978-0-07-890853-8

Answers for Workbooks The answers for Chapter 7 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the English workbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9) These masters contain a Spanish version of Chapter 7 Test Form 2A and Form 2C.

Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:

Glencoe/McGraw-Hill

8787 Orion Place

Columbus, OH 43240

ISBN13: 978-0-07-890516 - 2

ISBN10: 0-07-890516-8

Printed in the United States of America

1 2 3 4 5 6 7 8 9 10 009 14 13 12 11 10 09 08

iii

Contents

Teacher’s Guide to Using the Chapter 7

Resource Masters .............................................iv

Chapter ResourcesStudent-Built Glossary ....................................... 1

Anticipation Guide (English) .............................. 3

Anticipation Guide (Spanish) ............................. 4

Lesson 7-1Ratios and Proportions

Study Guide and Intervention ............................ 5

Skills Practice .................................................... 7

Practice .............................................................. 8

Word Problem Practice ..................................... 9

Enrichment ...................................................... 10

Graphing Calculator Activity ............................ 11

Lesson 7-2Similar Polygons

Study Guide and Intervention .......................... 12

Skills Practice .................................................. 14

Practice ............................................................ 15

Word Problem Practice ................................... 16

Enrichment ...................................................... 17

Lesson 7-3Similar Triangles


Skills Practice .................................................. 20

Practice ............................................................ 21


Enrichment ...................................................... 23

Lesson 7-4Parallel Lines and Proportional Parts


Skills Practice .................................................. 26

Practice ............................................................ 27


Enrichment ...................................................... 29

Lesson 7-5Parts of Similar Triangles


Skills Practice .................................................. 32

Practice ............................................................ 33


Enrichment ...................................................... 35

Spreadsheet Activity ........................................ 36

Lesson 7-6Similarity Transformations


Skills Practice .................................................. 39

Practice ............................................................ 40


Enrichment ...................................................... 42

Lesson 7-7Scale Drawings and Models


Skills Practice .................................................. 45

Practice ............................................................ 46


Enrichment ...................................................... 48

AssessmentStudent Recording Sheet ................................ 49

Rubric for Extended-Response ....................... 50

Chapter 7 Quizzes 1 and 2 ............................. 51

Chapter 7 Quizzes 3 and 4 ............................. 52

Chapter 7 Mid-Chapter Test ............................ 53

Chapter 7 Vocabulary Test ............................. 54

Chapter 7 Test, Form 1 ................................... 55

Chapter 7 Test, Form 2A ................................. 57

Chapter 7 Test, Form 2B ................................. 59

Chapter 7 Test, Form 2C ................................ 61

Chapter 7 Test, Form 2D ................................ 63

Chapter 7 Test, Form 3 ................................... 65

Chapter 7 Extended-Response Test ............... 67

Standardized Test Practice ............................. 68

Answers ........................................... A1–A32

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iv

Teacher’s Guide to Using the

Chapter 7 Resource MastersThe Chapter 7 Resource Masters includes the core materials needed for Chapter 7. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing on the

TeacherWorks PlusTM CD-ROM.

Chapter Resources

Student-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to recording definitions and/or examples for each term. You may suggest that student highlight or star the terms with which they are not familiar. Give to students before beginning Lesson 7-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 3–4) This master presented in both English and Spanish is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their perceptions have changed.

Lesson Resources

Study Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the

lesson.

Practice This master closely follows the types of problems found in the Exercises section of the Student Edition and includes word problems. Use as an additional practice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the mathematics they are learning. They are written for use with all levels of students.

Graphing Calculator, TI-Nspire, or

Spreadsheet Activities These activities present ways in which technology can be used with the concepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presentation.

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Assessment OptionsThe assessment masters in the Chapter 7

Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended-Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test provides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choice questions and is intended for use with below grade level students.

• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Form 3 is a free-response test for use with above grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Sample answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

• Full-size answer keys are provided for the assessment masters.

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NAME DATE PERIOD

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 7.

As you study the chapter, complete each term’s definition or description.

Remember to add the page number where you found the term. Add these pages to

your Geometry Study Notebook to review vocabulary at the end of the chapter.

(continued on the next page)

Chapter 7 1 Glencoe Geometry

Student-Built Glossary

Vocabulary TermFound

on PageDe! nition/Description/Example

cross products

dilation

enlargement

means

midsegment of a triangle

proportion

7

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Vocabulary TermFound

on PageDe! nition/Description/Example

ratio

reduction

scale drawing

scale factor

scale model

similar polygons

similarity transformation

Student-Built Glossary (continued)7

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Anticipation Guide

Proportions and Similarity

7

Before you begin Chapter 7

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 7

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

Step 2

Step 1

STEP 1A, D, or NS

StatementSTEP 2A or D

1. Ratios are always written as fractions.

2. A proportion is an equation stating that two ratios are equal.

3. Two ratios are in proportion to each other only if their cross products are equal.

4. If a −

b =

c −

d , then ad = bc.

5. The ratio of the lengths of the sides of similar figures is called the scale factor for the two figures.

6. If one angle in a triangle is congruent to an angle in another triangle, then the two triangles are similar.

7. If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then the line separates the two sides into congruent segments.

8. A segment whose endpoints are the midpoints of two sides of a triangle is parallel to the third side of the triangle.

9. If two triangles are similar then their perimeters are equal.

10. The medians of two similar triangles are in the same proportion as corresponding sides.

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NOMBRE FECHA PERÍODO

Antes de comenzar el Capítulo 7

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a).

Después de completar el Capítulo 7

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.

Ejercicios preparatorios

Proporciones y Semejanzas

Paso 1

PASO 1A, D o NS

EnunciadoPASO 2A o D

1. Las razones se escriben siempre como fracciones.

2. Una proporción es una ecuación que establece que dos razones son iguales.

3. Dos razones son proporcionales entre sí sólo si sus productos cruzados son iguales.

4. Si a −

b =

c −

d , entonces ad = bc.

5. La razón de las longitudes de los lados de figuras semejantes se llama factor de escala para las dos figuras.

6. Si un ángulo de un triángulo es congruente a un ángulo de otro triángulo, entonces los dos triángulos son semejantes.

7. Si una recta es paralela a un lado del triángulo e interseca los otros dos lados en dos puntos diferentes, entonces la recta separa los dos lados en segmentos congruentes.

8. Un segmento cuyos extremos son los puntos medios de dos lados de un triángulo es paralelo al tercer lado del triángulo.

9. Si dos triángulos son semejantes, éstos tienen el mismo perímetro.

10. Las medianas de dos triángulos semejantes están en la misma proporción que los lados correspondientes.

Paso 2

7

Capítulo 7 4 Geometría de Glencoe

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Lesso

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-1


Study Guide and Intervention

Ratios and Proportions

Write and Use Ratios A ratio is a comparison of two quantities by divisions. The ratio a to b, where b is not zero, can be written as

a −

b or a:b.

In 2007 the Boston RedSox baseball team won 96 games out of 162

games played. Write a ratio for the number of games won to the total number of

games played.

To find the ratio, divide the number of games won by the total number of games played. The

result is 96 −

162 , which is about 0.59. The Boston RedSox won about 59% of their games in

2007.

The ratio of the measures of the angles

in △JHK is 2:3:4. Find the measures of the angles.

The extended ratio 2:3:4 can be rewritten 2x:3x:4x.

Sketch and label the angle measures of the triangle. Then write and solve an equation to find the value of x.

2x + 3x + 4x = 180 Triangle Sum Theorem

9x = 180 Combine like terms.

x = 20 Divide each side by 9.

The measures of the angles are 2(20) or 40, 3(20) or 60, and 4(20) or 80.

Exercises

1. In the 2007 Major League Baseball season, Alex Rodriguez hit 54 home runs and was at bat 583 times. What is the ratio of home runs to the number of times he was at bat?

2. There are 182 girls in the sophomore class of 305 students. What is the ratio of girls to total students?

3. The length of a rectangle is 8 inches and its width is 5 inches. What is the ratio of length to width?

4. The ratio of the sides of a triangle is 8:15:17. Its perimeter is 480 inches. Find the length of

each side of the triangle.

5. The ratio of the measures of the three angles of a triangle is 7:9:20. Find the measure of each

angle of the triangle.

Example 1

Example 2

7-1

4x° 2x°

3x°

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Study Guide and Intervention (continued)


Use Properties of Proportions A statement that two ratios are

equal is called a proportion. In the proportion a − b = c −

d , where b and d are

not zero, the values a and d are the extremes and the values b and c

are the means. In a proportion, the product of the means is equal to the

product of the extremes, so ad = bc. This is the Cross Product Property.

Solve 9 −

16 =

27 −

x .

9 − 16

= 27 −

x

9 · x = 16 · 27 Cross Products Property

9x = 432 Multiply.

x = 48 Divide each side by 9.

POLITICS Mayor Hernandez conducted a random survey of

200 voters and found that 135 approve of the job she is doing. If there are

48,000 voters in Mayor Hernandez’s town, predict the total number of voters

who approve of the job she is doing.

Write and solve a proportion that compares the number of registered voters and the number of registered voters who approve of the job the mayor is doing.

135 −

200 = x

− 48,000

← voters who approve

← all voters

135 · 48,000 = 200 · x Cross Products Property

6,480,000 = 200x Simplify.

32,400 = x Divide each side by 200.

Based on the survey, about 32,400 registered voters approve of the job the mayor is doing.

Exercises

Solve each proportion.

1. 1 − 2 = 28

− x 2.

3 −

8 =

y −

24 3.

x + 22 −

x + 2 = 30

− 10

4. 3 −

18.2 = 9 − y 5.

2x + 3 −

8 = 5 −

4 6.

x + 1 −

x - 1 = 3 −

4

7. If 3 DVDs cost $44.85, find the cost of one DVD.

8. BOTANY Bryon is measuring plants in a field for a science project. Of the first 25 plants he measures, 15 of them are smaller than a foot in height. If there are 4000 plants in the field, predict the total number of plants smaller than a foot in height.

7-1

Example 1

Example 2

a − b = c −

d

a · d = b · c

extremes means↑ ↑

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Lesso

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Skills Practice


7-1

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

per game.

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?

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.

4. BOARD GAMES Myra is playing a board game. After 12 turns, Myra has landed on a

blue space 3 times. If the game will last for 100 turns, predict how many times Myra

will land on a blue space.

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, predict the number

of female students?


6. 2 − 5 =

x −

40 7.

7 −

10 = 21

− x 8.

20 −

5 =

4x −

6

9. 5x

− 4 =

35 −

8 10.

x+1 −

3 =

7 −

2 11.

15 −

3 =

x-3 −

5

12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is

450 centimeters. Find the measures of each side of the triangle.

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

What are the measures of the sides of the triangle?

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

What are the measures of the sides of the triangle?

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

Find the measures of each side of the triangle.

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Practice


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. 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?

3. QUALITY CONTROL A worker at an automobile assembly plant checks new cars for

defects. Of the first 280 cars he checks, 4 have defects. If 10,500 cars will be checked this

month, predict the total number of cars that will have defects.


4. 5 −

8 =

x −

12 5.

x −

1.12 = 1 −

5 6.

6x −

27 = 43

7. x + 2

− 3 =

8 −

9 8.

3x - 5 −

4 =

-5 −

7 9. x -2

− 4 =

x + 4 −

2

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

Find the measure of each side of the triangle.


84 inches. Find the measure of each side of the triangle.


77 centimeters. Find the measure of each side of the triangle.

13. The ratio of the measures of the three angles is 4:5:6. Find the measure of each angle of

the triangle.

14. The ratio of the measures of the three angles is 5:7:8. Find the measure of each angle of

the triangle.

15. BRIDGES A construction worker is placing rivets in a new bridge. He uses 42 rivets to build

the first 2 feet of the bridge. If the bridge is to be 2200 feet in length, predict the number of

rivets that will be needed for the entire bridge.

7-1

Lesso

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Word Problem Practice


1. TRIANGLES The ratios of the measures

of the angles in △DEF is 7:13:16.

13x°

16x°

7x°

Find the measure of the angles.

2. RATIONS Sixteen students went on

a week-long hiking trip. They brought

with them 320 specially baked, protein-

rich, cookies. What is the ratio of cookies

to students?

3. CLOVERS Nathaniel is searching for

a four-leaf clover in a field. He finds

2 four-leaf clovers during the first

12 minutes of his search. If Nathaniel

spends a total of 180 minutes searching

in the field, predict the number of four-

leaf clovers Nathaniel will find.

4. CARS A car company builds two

versions of one of its models—a sedan

and a station wagon. The ratio of sedans

to station wagons is 11:2. A freighter

begins unloading the cars at a dock.

Tom counts 18 station wagons and then

overhears a dock worker call out, “Okay,

that’s all of the wagons . . . bring out

the sedans!” How many sedans were on

the ship?

5. DISASTER READINESS The town of

Oyster Bay is conducting a survey of

80 households to see how prepared its

citizens are for a natural disaster. Of

those households surveyed, 66 have a

survival kit at home.

a. Write the ratio of people with

survival kits in the survey.

b. Write the ratio of people without

survival kits in the survey.

c. There are 29,000 households in

Oyster Bay. If the town wishes to

purchase survival kits for all

households that do not currently have

one, predict the number of kits it will

have to purchase.

7-1

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Enrichment

Growth Charts

It is said that when a child has reached the age of 2 years, he is roughly half of his adult

height. The growth chart below shows the growth according to percentiles for boys.

1. Use the chart to determine the approximate height for a boy at age 2 if he is in the 75th

to 95th percentile.

2. Using the rule that the height at age 2 is approximately half of his adult height, set up a

proportion to solve for the adult height of the boy in Exercise 1. Solve your proportion.

3. Use the chart to approximate the height at age 18 for a boy if he is in the 75th to 95th

percentile. How does this answer compare to the answer to problem 1?

4. Repeat this process for a boy who is in the 5th to 25th percentile.

5. Is using the rule that a boy is half of his adult height at age 2 years a good

approximation? Explain.

7-1

85

75

65

55

45

35

2 4 6 8 10 12 14 16 18

Age (yr)

Heig

ht

(in

.)

Percentile Group

5th to 25th

25th to 75th

75th to 95th

A

B

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Lesso

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-1


You can use a calculator to solve proportions.

Solve the proportion by using cross products. Round your answer

to the nearest hundredth.

55.6

− 16.9

= 45.8

− x

Multiply 16.9 by 45.8. Divide the product by 55.6.

Enter: 16.9 3 45.8 ÷ 55.6 ENTER 13.92122302

The solution is approximately 13.92

Solve each proportion by using cross products. Round your answers

to the nearest hundredth.

1. 13.9

− 39.8

= 10.5

− x 2.

25.9 −

37.7 =

24.3 −

x 3.

19.6 −

54.3 =

27.7 −

x

4. 66.8

− 43.4

= x −

16.9 5.

75.4 −

37.2 =

x −

32.4 6.

x −

46.2 =

29.7 −

36.4

7. x −

14.9 =

16.8 −

24.6 8.

35.8 −

x =

32.9 −

27.8 9.

46.9 −

x =

15.7 −

99.9

10. 34.9

− 21.1

= x −

36.6 11.

x −

99.8 =

32.2 −

45.3 12.

68.9 −

x =

44.3 −

86.4

7-1 Graphing Calculator Activity

Solving Proportions

Example

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Identify Similar Polygons Similar polygons have the same shape but not necessarily the same size.

If △ABC ∼ △XYZ, list all pairs of congruent angles and

write a proportion that relates the corresponding sides.

Use the similarity statement.

Congruent angles: ∠A ! ∠X, ∠B ! ∠Y, ∠C ! ∠Z

Proportion: AB −

XY =

BC −

YZ =

CA −

ZX

Determine whether the pair of figures is similar. If so, write the

similarity statement and scale factor. Explain your reasoning.

Step 1 Compare corresponding angles.

∠W ! ∠P, ∠X ! ∠Q, ∠Y ! ∠R, ∠Z ! ∠S

Corresponding angles are congruent.

Step 2 Compare corresponding sides.

WX

− PQ

= 12

− 8 =

3 −

2 , XY

− QR

= 18

− 12

= 3 −

2 , YZ

− RS

= 15

− 10

= 3 −

2 , and

ZW

− SP

= 9 −

6 =

3 −

2 . Since corresponding sides are proportional,

WXYZ ∼ PQRS. The polygons are similar with a scale factor of 3 −

2 .

Exercises

List all pairs of congruent angles, and write a

proportion that relates the corresponding sides

for each pair of similar polygons.

1. △DEF ∼ △GHJ

2. PQRS ∼ TUWX

Determine whether each pair of figures is similar. If so, write the similarity

statement and scale factor. If not, explain your reasoning.

3. 4.

22

11

7-2 Study Guide and Intervention

Similar Polygons

Example 1

15

9

12

18

XW

ZY

10

6

8

12

QP

S R

Example 2

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Similar Polygons

Use Similar Figures You can use scale factors and proportions to find missing side lengths in similar polygons.

The two polygons are

similar. Find x and y.

x

y38

32

16 13

T

S

R P

N

M

Use the congruent angles to write the corresponding vertices in order.

△RST ∼ △MNP

Write proportions to find x and y.

32

− 16

= x −

13

38 − y =

32 −

16

16x = 32(13) 32y = 38(16)

x = 26 y = 19

If △DEF ∼ △GHJ, find the

scale factor of △DEF to △GHJ and the

perimeter of each triangle.

The scale factor is

EF −

HJ =

8 −

12 = 2 −

3 .

The perimeter of △DEF is 10 + 8 + 12 or 30.

2 − 3 =

Perimeter of △DEF −−

Perimeter of △GHJ Theorem 7.1

2 − 3 =

30 − x Substitution

(3)(30) = 2x Cross Products Property

45 = x Solve.

So, the perimeter of △GHJ is 45.

Exercises

Each pair of polygons is similar. Find the value of x.

1.

9

12

8

10x

2.

10

4.5x

2.5

5

3. 12

x + 1

18

24

18

4.

40

36x +15

30

18

5. If ABCD ∼ PQRS, find the scale factor

of ABCD to PQRS and the perimeter

of each polygon.

7-2

Example 1 Example 2

12

12

10

8

14

20

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Determine whether each pair of figures is similar. If so, write the similarity


1. 2.

7.5

7.5

7.5

7.5

Z Y

XW

3

3

33

S R

QP


3. 4.

5. 6.

9

10

3

x + 1

5

P

S

R

U V

WT

Q

10

8

x + 2

x - 1

SP

M N

L

T

x + 5

x + 5

9

3

44S

W

XY

T

U26

14

12 13

B C

DA

13

7

6 x

G

E H

F

10.5

6 9

59° 35°A C

B

7

4 6

59° 35°D F

E

Skills Practice

Similar Polygons

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Practice

Similar PolygonsDetermine whether each pair of figures is similar. If so, write the similarity


1.

25 12

9

14.4

15

2420

15

JM Q R

S

PKL

2.

2412

14 162118

VUAC

BT


3.

14

10

x + 6

x + 9

C N MD

A BLP

4.

6 12

40°

40°

x + 1

x - 3A

F

E

D

B C

5. PENTAGONS If ABCDE ∼ PQRST, find the

scale factor of ABCDE to PQRST and the

perimeter of each polygon.

6. SWIMMING POOLS The Minnitte family

and the neighboring Gaudet family both

have in-ground swimming pools. The

Minnitte family pool, PQRS, measures

48 feet by 84 feet. The Gaudet family pool,

WXYZ, measures 40 feet by 70 feet. Are

the two pools similar? If so, write the

similarity statement and scale factor.

7-2

2015

10

21

48 ft

84 ft

70 ft

40 ftP

Q

S W

Z

X

Y

R

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Similar Polygons

1. PANELS When closed, an entertainment

center is made of four square panels.

The three smaller panels are congruent

squares.

What is the scale factor of the larger

square to one of the smaller squares?

2. WIDESCREEN TELEVISIONS An

electronics company manufactures

widescreen television sets in several

different sizes. The rectangular viewing

area of each television size is similar to

the viewing areas of the other sizes. The

company’s 42-inch widescreen television

has a viewing area perimeter of

approximately 144.4 inches. What is the

viewing area perimeter of the company’s

46-inch widescreen television?

3. ICE HOCKEY An official Olympic-sized

ice hockey rink measures 30 meters by

60 meters. The ice hockey rink at the

local community college measures 25.5

meters by 51 meters. Are the ice hockey

rinks similar? Explain your reasoning.

4. ENLARGING Camille wants to make

a pattern for a four-pointed star with

dimensions twice as long as the one

shown. Help her by drawing a star

with dimensions twice as long on the

grid below.

5. BIOLOGY A paramecium is a small

single-cell organism. The paramecium

magnified below is actually one tenth

of a millimeter long.

a. If you want to make a photograph of

the original paramecium so that its

image is 1 centimeter long, by what

scale factor should you magnify it?

b. If you want to make a photograph of

the original paramecium so that its

image is 15 centimeters long, by what

scale factor should you magnify it?

c. By approximately what scale factor

has the paramecium been enlarged

to make the image shown?

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Enrichment

Constructing Similar PolygonsHere are four steps for constructing a polygon that is similar to and with sides twice as long as those of an existing polygon.

Step 1 Choose any point either inside or outside the polygon and label it O.

Step 2 Draw rays from O through each vertex of the polygon.

Step 3 For vertex V, set the compass to length OV. Then locate a new point V′ on ray OV such that VV′ = OV. Thus, OV′ = 2(OV).

Step 4 Repeat Step 3 for each vertex. Connect points V′, W′, X′ and Y′ to form the new polygon.

Two constructions of polygons similar to and with sides twice those of VWXY are shown below. Notice that the placement of point O does not affect the size or shape of V′W′X′Y′, only its location.

VW

Y X

VW

Y X

V'

W'

X'Y'

O

VW

YX

V'

W'

X'Y'

O

Trace each polygon. Then construct a similar polygon with sides twice as long as

those of the given polygon.

1.

C

A

B

2. DE

F

G

3. Explain how to construct a similar polygon with sides three times the length of those of polygon HIJKL. Then do the construction.

4. Explain how to construct a similar

polygon 1 1 − 2 times the length of those of

polygon MNPQRS. Then do the construction.

7-2

M

S

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Similar Triangles

Identify Similar Triangles Here are three ways to show that two triangles are similar.

AA Similarity Two angles of one triangle are congruent to two angles of another triangle.

SSS Similarity The measures of the corresponding side lengths of two triangles are proportional.

SAS SimilarityThe measures of two side lengths of one triangle are proportional to the measures of two

corresponding side lengths of another triangle, and the included angles are congruent.

Determine whether the

triangles are similar.

15

129

D E

F

10

86

A B

C

AC −

DF =

6 −

9 = 2 −

3

BC −

EF =

8 −

12 = 2 −

3

AB −

DE =

10 −

15 = 2 −

3

△ABC ∼ △DEF by SSS Similarity.

Determine whether the


8

4

70°

Q

R S6

3

70°

M

N P

3 −

4 =

6 −

8 , so

MN −

QR =

NP −

RS .

m∠N = m∠R, so ∠N & ∠R.

△NMP ∼ △RQS by SAS Similarity.

Exercises

Determine whether the triangles are similar. If so, write a similarity statement.

Explain your reasoning.

1.

A

B

C

D

E

F

2.

36

2018

24

J

K

L

W X

Y

3. RS

UL

T

4.

18

8

65°9

465°

D

K J

P

ZR

5.

39

26

1624

F

B E

L

MN

6. 32

20

154025

24

G H

Q R

SI

7-3

Example 1 Example 2

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Similar Triangles

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

△ABC ∼ △DEF. Find the

values of x and y.

18√3

B

CA

18 189

y

x

E

FD

AC

− DF

= BC

− EF

AB −

DE =

BC −

EF

18 √ $ 3

− x =

18 −

9

y −

18 =

18 −

9

18x = 9(18 √ $ 3 ) 9y = 324

x = 9 √ $ 3 y = 36

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?

7 ft

6 ft

?

1.5 ft

The sun’s rays form similar triangles.

Using x for the height of the pole, 6 −

x =

1.5 −

7 ,

so 1.5x = 42 and x = 28.

The flagpole is 28 feet tall.

Exercises

ALGEBRA Identify the similar triangles. Then find each measure.

1. JL 2. IU

3. QR 4. BC

5. LM 6. QP

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?

x32

10

2230

PQ

N

R

F

x

9

8

7.2

16

K

VP M

L

24

36√2x

3636

Q V

R

S

T

35

13

10 20

J L

K

Y

X Zx

x

60

38.6

23

30

E

BA

C D

20

13

26

I

UW

G

H

x

7-3

Example 1 Example 2

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Determine whether each pair of triangles is similar. If so, write a similarity

statement. If not, what would be sufficient to prove the triangles similar? Explain

your reasoning.

1. 2.

3. 4.

21

70°

15

MJ

P

1470°10

U

S

T

30°60°RQ

M

S

K

T


5. AC 6. JL

15

12

D

E CB

Ax + 1

x + 5

16

4

M

NL

J

K

x + 18

x - 3

7. EH 8. VT

12

9

6

9

D

E

FH

Gx + 5

6

14

S

U

V

R T

3x - 3

x + 2

12

128

9

96

C P

QR

A

B

Skills Practice

Similar Triangles

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Practice

Similar Triangles

Determine whether the triangles are similar. If so, write a similarity statement. If

not, what would be sufficient to prove the triangles similar? Explain your

reasoning.

1. 2.


3. LM, QP 4. NL, ML

5. 6.

8

N

J LK

Mx + 5

6x + 2

24

12

18

N

P

QL

M

x + 3

x - 1

42°

42°

24

1812

16J

A K

Y S

W

7-3

105

12

8

86

x + 7x - 1

x + 1

x + 33

4

7. INDIRECT MEASUREMENT A lighthouse casts a 128-foot shadow. A nearby lamppost

that measures 5 feet 3 inches casts an 8-foot shadow.

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

b. What is the height of the lighthouse?

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Similar Triangles

A

B

D

E

C

1. CHAIRS A local furniture store sells two

versions of the same chair: one for

adults, and one for children. Find the

value of x such that the chairs are

similar.

18˝

16˝

X

12˝

2. BOATING The two sailboats shown are

participating in a regatta. Find the

value of x.

168.5 in. 220 in.

X in. 264 in.

40°40°

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

△CEB. They look similar to her. Prove

that this is the case.

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?

5. MOUNTAIN PEAKS 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.

Gavin

Brianna

Peak

2 in.

2.015 in.

0.246 in.90˚

83˚

The actual distance between Gavin and

Brianna’s houses is 1 1−2

miles.

a. What is the actual distance of the

mountain peak from Gavin’s house?

Round your answer to the nearest

tenth of a mile.

b. What is the actual distance of the

mountain peak from Brianna’s house?


tenth of a mile.

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Moving Shadows

Have you ever watched your shadow as you

walked along the street at night and observed

how its shape changes as you move? Suppose

a man who is 6 feet tall is standing below

a lamppost that is 20 feet tall. The man is

walking away from the lamppost at a rate of

5 feet per second.

1. If the man is moving at a rate of 5 feet per

second, make a conjecture as to the rate

that his shadow is moving.

2. How far away from the lamppost is the man after 8 seconds?

3. How far is the end of his shadow from the bottom of the lamppost after

8 seconds? Use similar triangles to solve this problem.

4. After 3 more seconds, how far from the lamppost is the man? How far from the lamppost

is his shadow?

5. How many feet did the man move in 3 seconds? How many feet did the shadow move in

3 seconds?

6. The man is moving at a rate of 5 feet/second. What rate is his shadow moving? How

does this rate compare to the conjecture you made in Exercise 1? Make a conjecture as to

why the results are like this.

20 ft

6 ft

x ft

5 ft/sec

Enrichment7-3

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Parallel Lines and Proportional Parts

Proportional Parts within Triangles In any triangle, a line parallel to one side of a triangle separates the other two sides proportionally. This is the Triangle Proportionality Theorem.The converse is also true.

If "# XY ‖ "# RS , then RX

− XT

= SY

− YT

. If RX

− XT

= SY

− YT

, then "# XY ‖ "# RS .

If X and Y are the midpoints of −− RT and

−− ST , then −− XY is a

midsegment of the triangle. The Triangle Midsegment Theorem

states that a midsegment is parallel to the third side and is half its length.

If −− XY is a midsegment, then "# XY ‖ "# RS and XY = 1 −

2 RS.

YS

T

R

X

In △ABC, −−

EF ‖ −−

CB . Find x.

F

C

BA

18

x + 22 x + 2

6E

Since −− EF ‖

−−− CB , AF −

FB = AE

− EC

.

x + 22

− x + 2

= 18

− 6

6x + 132 = 18x + 36

96 = 12x

8 = x

In △GHJ, HK = 5,

KG = 10, and JL is one-half the length

of −−

LG . Is −−−

HK ‖ −− KL ?

Using the converse of the Triangle

Proportionality Theorem, show that

HK −

KG =

JL −

LG .

Let JL = x and LG = 2 x.

HK −

KG = 5 −

10 = 1 −

2

JL −

LG = x −

2x = 1 −

2

Since 1 − 2 = 1 −

2 , the sides are proportional and

−−− HJ ‖ −− KL .

Exercises

ALGEBRA Find the value of x.

1. 5

5 x

7 2.

9

20

x

18 3.

35

x

4.

30 10

24

x 5. 11

x + 12

x

33

6.

30 10

x + 10

x

7-4

Example 1 Example 2

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Proportional Parts with Parallel Lines When three or more parallel lines cut two transversals, they separate the transversals into proportional parts. If the ratio of the parts is 1, then the parallel lines separate the transversals into congruent parts.

If ℓ1 ‖ ℓ

2 ‖ ℓ

3, If ℓ

4 ‖ ℓ

5 ‖ ℓ

6 and

then a −

b =

c −

d .

u − v = 1, then

w − x = 1.

Refer to lines ℓ1, ℓ

2, and ℓ

3 above. If a = 3, b = 8, and c = 5, find d.

ℓ1 ‖ ℓ

2 ‖ ℓ

3 so

3 −

8 =

5 −

d . Then 3d = 40 and d = 13 1 −

3 .

Exercises

ALGEBRA Find x and y.

1.

x + 12 12

3x5x

2. 2x - 6

x + 3

12

3.

2x + 4

3x - 1 2y + 2

3y

4.

5

8 y + 2

y

5. 3 4

x + 4x

6.

1632

y + 3y

a

bd

u v

xw

ct

n

m

s ℓ1

ℓ4 ℓ5 ℓ6

ℓ2

ℓ3

7-4

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1. If JK = 7, KH = 21, and JL = 6, 2. If RU = 8, US = 14, TV = x - 1,

find LI. and VS = 17.5, find x and TV.

Determine whether −−

BC ‖ −−

DE . Justify your answer.

3. AD = 15, DB = 12, AE = 10, and EC = 8

4. BD = 9, BA = 27, and CE = 1 −

3 EA

5. AE = 30, AC = 45, and AD = 2DB

−−

JH is a midsegment of △KLM. Find the value of x.

6.

30

x

7.

9

x

8.

8

x

ALGEBRA Find x and y.

9.

2x + 1 3y - 8

y + 5x + 7

10.

3–2x + 2

x + 3

2y - 1

3y - 5

C

B

E

D

A

R

U V

T

S

HL

K J

I

Skills Practice


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Practice


1. If AD = 24, DB = 27, and EB = 18, 2. If QT = x + 6, SR = 12, PS = 27, and

find CE. TR = x - 4, find QT and TR.

Determine whether −−

JK ‖ −−−

NM . Justify your answer.

3. JN = 18, JL = 30, KM = 21, and ML = 35

4. KM = 24, KL = 44, and NL = 5 −

6 JN

−−

JH is a midsegment of △KLM. Find the value of x.

5.

22x

6.

x + 7

x

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

3x - 42–3y + 3

1–3y + 65

–4x + 3

4–3y + 2

3x - 4 x + 1

4y - 4

9. MAPS On a map, Wilmington Street, Beech Drive, and Ash Grove

Lane appear to all be parallel. 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?

Ash Grove

Beech

Magnolia Kendall

Wilmington

NLJ

KM

S

Q

T

RP

D

E

C

BA

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Meeeeowww!

30 ft

40 ft

46 ft

x

8 ft 8 ft

10 ft

24 ft

10 ft

x

Cay St.

1 km 1.4 km

0.8 km

Bay St.

Dal

e S

t.

Earl St.

(not drawn to scale)

x

1. CARPENTRY Jake is

fixing an A-frame.

He wants to add a

horizontal support

beam halfway up

and parallel to

the ground. How

long should this

beam be?

2. STREETS In the diagram, Cay Street

and Bay Street are parallel. Find x.

3. JUNGLE GYMS Prassad is building a

two-story jungle gym according to the

plans shown. Find x.

4. FIREMEN A cat

is stuck in a tree

and firemen try to

rescue it. Based

on the figure, if a

fireman climbs

to the top of the

ladder, how far

away is the cat?

5. EQUAL PARTS Nick has a stick that he

would like to divide into 9 equal parts.

He places it on a piece of grid paper as

shown. The grid paper is ruled so that

vertical and horizontal lines are equally

spaced.

a. Explain how he can use the grid

paper to help him find where he

needs to cut the stick.

b. Suppose Nick wants to divide his

stick into 5 equal parts utilizing the

grid paper. What can he do?

8 feet

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Enrichment

Parallel Lines and Congruent Parts

There is a theorem stating that if three parallel lines cut off congruent segments on one

transversal, then they cut off congruent segments on any transversal. This can be shown

for any number of parallel lines. The following drafting technique uses this fact to divide a

segment into congruent parts.

−− AB to be separated into five congruent parts. This can be

done very accurately without using a ruler. All that is needed

is a compass and a piece of notebook paper.

Step 1 Hold the corner of a piece of notebook paper at

point A.

Step 2 From point A, draw a segment along the paper

that is five spaces long. Mark where the lines

of the notebook paper meet the segment. Label

the fifth point P.

Step 3 Draw −−

PB . Through each of the other marks

on −−

AP , construct a line parallel to −−

BP . The

points where these lines intersect −−

AB will

divide −−

AB into five congruent segments.

Use a compass and a piece of notebook paper to divide

each segment into the given number of congruent parts.

1. six congruent parts

2. seven congruent parts A B

A B

A

P

B

A B

P

A B

A B

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7-5 Study Guide and Intervention

Parts of Similar Triangles

Special Segments of Similar Triangles When two triangles are similar, corresponding altitudes, angle bisectors, and medians are proportional to the corresponding sides.

Exercises

Find x.

1. x

36

1820

2.

69

12x

3.

3

3

4

x

4.

10

10x

8

87

5.

45

42

x

30 6.

14

12

12

x

Example 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.

AB −

XY = BD

− WY

24 −

x =

10 −

8

10x = 24(8)

10x = 192

x = 19.2

24

10

DW

Y

ZXC

B

A

8

x

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7-5 Study Guide and Intervention (continued)


Triangle Angle Bisector Theorem An angle bisector in a triangle separates the opposite side into two segments that are proportional to the lengths of the other two sides.

Find x.

Since −−−

SU is an angle bisector, RU

− TU

= RS

− TS

.

x −

20 =

15 −

30

30x = 20(15)

30x = 300

x = 10

Exercises

Find the value of each variable.

1. 25

20 28

x 2.

15 9

3a

3.

36 42

46.8

n

4.

10

15

13x

5.

110

100

160

r

6.

17

11

x + 7

x

Example

x 20

3015

UT

S

R

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7-5 Skills Practice

Parts of Similar TrianglesFind x.

1. 2.

7

7

22

x5.25

5.25

15

33

10

x

3. 4.

20

10

x

22

18

12

12.6

x

5. If △RST ∼ △EFG, −−−

SH is an 6. If △ABC ∼ △MNP, −−−

AD is an

altitude of △RST, −−

FJ is an altitude of altitude of △ABC, −−−

MQ is an altitude of

△EFG, ST = 6, SH = 5, and FJ = 7, △MNP, AB = 24, AD = 14, and

find FG. MQ = 10.5, find MN.

JGE

F

HTR

S

C

A

BD PNQ

M

Find the value of each variable.

7. 8.

26

128

m

7

20

24

x

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-5


Practice


ALGEBRA Find x.

1.

3230

24x

2.

26

25

39

x

3.

402x + 1

25x + 4

4.

5. If △JKL ∼ △NPR, −−−KM is an 6. If △STU ∼ △XYZ,

−−−UA is an

altitude of △JKL, −−PT is an altitude altitude of △STU,

−−ZB is an altitude

of △NPR, KL = 28, KM = 18, and of △XYZ, UT = 8.5, UA = 6, and PT = 15.75, find PR. ZB = 11.4, find ZY.

MJ L

K

TN R

P

YX B

Z

TS A

U

7. PHOTOGRAPHY Francine has a camera in which the distance from the lens to the film is 24 millimeters.

a. 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.)

b. 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?

7-5

x

28

20 30

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1. FLAGS An oceanliner is flying two

similar triangular flags on a flag pole.

The altitude of the larger flag is three

times the altitude of the smaller flag. If

the measure of a leg on the larger flag is

45 inches, find the measure of the

corresponding leg on the smaller flag.

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?

x

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.

A

B

45˚

30 ft

40 ft

50 ft

How much farther from Hank is point B

versus point A?

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.

AB

C

DF

E

The rigging is done so that AE = EF,

AC = CD, and AB = BC. What is the

ratio of CF to BE?

5. COPIES 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.

39 mm

Media

n

Altitude

30 mm

29 mm

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.

a. What is the length of the drawn

altitude of the enlarged triangle?


millimeter.

b. What is the length of the drawn

median of the enlarged triangle?


millimeter.

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A Proof of Pythagorean Theorem

1. For right triangle ABC with right angle C, and

altitude −−−

CD as shown at the right, name three

similar triangles.

2. List the three similar triangles as headings in

the table below. Use the figure to complete the

table to list the corresponding parts of the three

similar right triangles.

Short Leg

Long Leg

Hypotenuse

3. Use the corresponding parts of these similar triangles and their proportions

to complete the statements in the proof and algebraically prove the

Pythagorean Theorem.

Statements Reasons

1. Right triangle ABC with 1. Given

altitude −−−

CD .

2. 2. Corresponding parts of similar

triangles are in the same ratio.

3. 3. Cross Products Property

4. 4. Addition Property of Equality

5. 5. Substitution

6. 6. Distributive Property

7. 7. Segment addition

8. a2 + b2 = c2 8. Substitution

A

a bh

d e

c

B

C

D

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You can use a spreadsheet to create a Sierpinski triangle.

Use a spreadsheet to create a Sierpinski triangle.

Step 1 In cell A2, enter 1. In cell A3, enter an equals sign followed by A2 + 1 . This will return the number of iterations. Click on the bottom right corner of cell A2 and drag it through cell A500 to get 500 iterations.

Step 2 In cell B1, enter 0.5. This indicates the midpoints of the segments.

Step 3 In cell C1, enter 1 and in cell D1, enter 0.3.

Step 4 In cell B2, enter an equals sign followed by 3*RAND(). Click on the bottom right corner of cell B2 and drag in through cell B500 to get 500 iterations.

Step 5 In cell C2, enter an equals sign followed by the recursive formula IF(B2<1,(12$B$1)*C1,IF(B2<2,(1-$B$1)*C1+$B$1,(1-$B$1)*C1+2*$B$1)). This will return the x values of the points to be graphed. Click on the bottom right corner of cell C2 and drag through cell C500.

Step 6 In cell D2, enter an equals sign followed by the recursive formula IF(B2<1,(1-$B$1)*D1,IF(B2<2,(1-$B$1)*D1+$B$1,(1-$B$1)*D1)). This will return the y values of the points to be graphed. Click on the bottom right corner of cell D2 and drag through cell D500.

Step 7 To graph the values in columns C and D, first highlight all of the data in the two columns. Next, choose the chart wizard from the toolbar. Select XY (Scatter). Press Next, Next. Then select the Gridlines tab and uncheck the Major gridlines. Then press Next and Finish. This will return the Sierpinski triangle.

Exercises

Analyze your drawing.

1. What happens to your drawing if you have more iterations? Try 1000, 2000, and 5000.

2. Change the 0.5 to 2/3 in cell B1. (Hint: You may need to enter 0.666666 for 2/3.) How does this change the picture?

3. Change 0.3 to 0.6 in cell D1. How does this change the drawing?

4. Enter the following into a new spreadsheet and describe what you see. Step 1: In cell A2, enter the formula = A1. Step 2: In cell B2, enter the formula = A1 + B1. Step 3: Click on the bottom right corner of cell B2 and drag through cell L2. Step 4: First, click on the 2 next to cell A2. Then, click on the bottom left corner of 2

and drag down through cell 12. Step 5: In cell A1, enter a 1 and press ENTER.

Sheet 1 Sheet 2 Sheet 3

Spreadsheet Activity

Fractals

7-5

Example

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7-6

Identify Similarity Transformations A dilation is a transformation that enlarges or reduces the original figure proportionally. The scale factor of a dilation, k, is the ratio of a length on the image to a corresponding length on the preimage. A dilation with k > 1 is an enlargement. A dilation with 0 < k < 1 is a reduction.

Determine whether the dilation from A to B is an enlargement or a

reduction. Then find the scale factor of the dilation.

Exercises

Determine whether the dilation from A to B is an enlargement or a reduction.

Then find the scale factor of the dilation.

1. y

x

2. y

x

3. y

x

4. y

x


Similarity Transformations

Example

a. y

x

B is larger than A, so the dilation is an enlargement.

The distance between the vertices at (-3, 4) and (-1, 4) for A is 2. The distance between the vertices at (0, 3) and (4, 3) for B is 4.

The scale factor is 4 − 2 or 2.

b. y

x

B is smaller than A, so the dilation is a reduction.

The distance between the vertices at (2, 3) and (2, -3) for A is 6. The distance between the vertices at (2, 1) and (2, -2) for B is 3.

The scale factor is 3 −

6 or 1 −

2 .

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Verify Similarity You can verify that a dilation produces a similar figure by comparing the ratios of all corresponding sides. For triangles, you can also use SAS Similarity.

Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation.

Example

a. original: A(-3, 4), B(2, 4), C(-3, -4)

image: D(1, 0), E(3.5, 0), F(1, -4)

Graph each figure. Since ∠A and ∠D are both right angles, ∠A " ∠D. Show that the lengths of the sides that include ∠A and ∠D are proportional to prove similarity by SAS.

Use the coordinate grid to find the lengths of vertical segments AC and DF and horizontal segments AB and DE.

AC

− DF

= 8 −

4 = 2 and AB

− DE

= 5 −

2.5 = 2,

so AC

− DF

= AB −

DE .

Since the lengths of the sides that include ∠A and ∠D are proportional, △ABC ∼ △DEF by SAS similarity.

b. original: G(-4, 1), H(0, 4), J(4, 1), K(0, -2)

image: L(-2, 1.5), M(0, 3), N(2, 1.5), P(0, 0)

Use the distance formula to find the length of each side.

GH = √ ''' 42 + 32 = √ '' 25 = 5

HJ = √ ''' 42 + 32 = √ '' 25 = 5

JK = √ ''' 42 + 32 = √ '' 25 = 5

KG = √ ''' 42 + 32 = √ '' 25 = 5

LM = √ '''' 22 + 1.52 = √ '' 6.25 = 2.5

MN = √ '''' 22 + 1.52 = √ '' 6.25 = 2.5

NP = √ '''' 22 + 1.52 = √ '' 6.25 = 2.5

PL = √ '''' 22 + 1.52 = √ '' 6.25 = 2.5

Find and compare the ratios of corresponding sides.

GH

− LM

= 5 −

2.5 = 2,

HJ −

MN =

5 −

2.5 = 2,

JK

− NP

= 5 −

2.5 = 2,

KC −

PL = 5

− 2.5

= 2.

Since GH

− LM

= HJ

− MN

= JK −

NP = KC

− PL

, GHJK ∼ LMNP.

y

x

y

x

Exercises

Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation.

1. A(-4, -3), B(2, 5), C(2, -3), 2. P(-4, 1), Q(-2, 4), R(0, 1), S(-2, -2),

D(-2, -2), E(1, 3), F(1, -2) W(1, -1.5), X(2, 0), Y(3, -1.5), Z(2, -3)

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7-6

Determine whether the dilation from A to B is an enlargement or

a reduction. Then find the scale factor of the dilation.

1. y

x

2. y

x

3. y

x

4. y

x

Graph the original figure and its dilated image. Then verify that the dilation is a

similarity transformation.

5. A(–3, 4), B(3, 4), C(–3, –2); 6. F(–3, 4), G(2, 4), H(2, –2), J(–3, –2);

A'(–2, 3), B'(0, 3), C'(–2, 1) F'(–1.5, 3), G'(1, 3), H'(1, 0), J'(–1.5, 0)

Skills Practice


y

x

y

x

7. P(–3, 1), Q(–1, 1), R(–1, –3); 8. A(–5, –1), B(0, 1), C(5, –1), D(0, –3);

P'(–1, 4), Q'(3, 4), R'(3, –4) A'(1, –1.5), B'(2, 0), C'(3, –1.5), D'(2, –3)

y

x

y

x

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7-6 Practice

Similarity TransformationsDetermine whether the dilation from A to B is an enlargement or a reduction.

Then find the scale factor of the dilation.

1. y

x

2. y

x

3. y

x

4. y

x

5. y

x

6. y

x

Graph the original figure and its dilated image. Then verify that the dilation is a

similarity transformation.

7. Q(1, 4), R(4, 4), S(4, -1), 8. A(-4, 2), B(0, 4), C(4, 2), D(0, -2),

X(-4, 5), Y(2, 5), Z(2, -5) F(-2, 1), G(0, 2), H(2, 1), J(0, -1)

9. FABRIC Ryan buys an 8-foot-long by 6-foot-wide piece of fabric as shown. He wants to cut a smaller, similar rectangular piece that has a scale factor of k = 1 −

4 . If point A(-4, 3)

is the top left-hand vertex of both the original piece of fabric and the piece Ryan wishes to cut out, what are the coordinates of the vertices for the piece Ryan will cut?

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7-6

1. CITY PLANNING The standard size of a city block in Manhattan is 264 feet by 900 feet. The city planner of Mechlinburg wants to build a new subdivision using similar blocks so the dimensions of a standard Manhattan block are enlarged by 2.5 times. What will be the new dimensions of each enlarged block?

2. STORAGE SHED A local home improvement store sells different sizes of storage sheds. The most expensive shed has a footprint that is 15 feet wide by 21 feet long. The least expensive shed has a footprint that is 10 feet wide by 14 feet long. Are the footprints of the two sheds similar? If so, tell whether the footprints of the least expensive shed is an enlargement or a reduction, and find the scale factor from the most expensive shed to the least expensive shed.

3. FIND THE ERROR Jeremy and Elisa are constructing dilations of rectangle ABCDfor their geometry class. Jeremy draws an enlargement FGHJ that contains the points F (-3, 3) and J(3, -6). Elisa draws a reduction KLMN that contains the points L(-3, 3) and N(-2, 2). Which person made an error in their dilation? Explain.

y

x

4. BANNERS The Bayside High School Spirit Squad is making a banner to take to away games. The banner they use for home games is shown below. If the new banner is to be a reduction of the home

game banner with a scale factor of 1−2

,

what will be the height of the new banner?

13 ft

3.5 ft

5. REASONING Consider the image QRSTof a rectangle WXYZ.

a. Is it possible that point Q and point Wcould have the same coordinates? If so, what must be true about point Q?

b. Is it possible that both points R and Xcould have the same coordinates if points T and Z have the same coordinates? If so, what are the possible values of the scale factor k for the dilation?



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7-6 Enrichment

Medial and Orthic TrianglesThe medial triangle is the triangle formed by connecting the

midpoints of each side of the triangle. The triangle formed by

A’, B’, and C’ is the medial triangle of triangle ABC.

Use a ruler and compass. Draw the medial triangle for each triangle below.

1. 2.

3. Use the triangles you have created to show that the medial triangle is similar to the

original triangle.

The orthic triangle is the triangle formed by connecting the

endpoints of each of the altitudes of the triangle. The triangle

formed by F, D, and, E is the orthic triangle of triangle ABC.

Use a ruler and compass. Draw the orthic triangle for each triangle below.

1. 2.

3. Use the triangles you have created to show that the orthic triangle is similar to the

original triangle.

'

''

'

''

'

''

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-7


7-7

Scale Models A scale model or a scale drawing is an object or drawing with lengths proportional to the object it represents. The scale of a model or drawing is the ratio of the length of the model or drawing to the actual length of the object being modeled or drawn.

MAPS The scale on the map shown is 0.75 inches : 6 miles. Find the

actual distance from Pineham to Menlo Fields.

Use a ruler. The distance between Pineham and

Menlo Fields is about 1.25 inches.

Method 1: Write and solve a proportion.

Let x represent the distance between cities.

0.75 in.

− 6 mi

= 1.25 in.

− x mi

0.75 · x = 6 · 1.25 Cross Products Property

x = 10 Simplify.

Method 2: Write and solve an equation.

Let a = actual distance and m = map distance

in inches. Write the scale as 6 mi −

0.75 in. ,

which is 6 ÷ 0.75 or 8 miles per inch.

a = 8 · m Write an equation.

= 8 · 1.25 m = 1.25 in.

= 10 Solve.

The distance between Pineham and Menlo Fields is 10 miles.

Example

map

actual

Exercises

Use the map above and a customary ruler to find the actual distance between

each pair of cities. Measure to the nearest sixteenth of an inch.

1. Eastwich and Needham Beach

2. North Park and Menlo Fields

3. North Park and Eastwich

4. Denville and Pineham

5. Pineham and Eastwich


Scale Drawings and Models

Denville

Pineham

Eastwich

North Park

Needham

Beach

Menlo Fields

0.75 in.

6 mi.

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7-7

Use Scale Factors The scale factor of a drawing or scale model is the scale written as a unitless ratio in simplest form. Scale factors are always written so that the model length in the ratio comes first.

SCALE MODEL A doll house that is 15 inches tall is a scale model of

a real house with a height of 20 feet.

a. What is the scale of the model?

To find the scale, write the ratio of a model length to an actual length.

model length

− actual length

= 15 in.

− 20 ft

or 3 in.

− 4 ft

The scale of the model is 3 in.:4 ft

b. How many times as tall as the actual house is the model?

Multiply the scale factor of the model by a conversion factor that relates inches to feet to

obtain a unitless ratio.

3 in.

− 4 ft

= 3 in.

− 4 ft

· 1 ft −

12 in. =

3 −

48 or 1 −

16

The scale factor is 1:16. That is, the model is 1 − 16

as tall as the actual house.

Exercises

1. MODEL TRAIN The length of a model train is 18 inches. It is a scale model of a train

that is 48 feet long. Find the scale factor.

2. 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.

3. 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?

Example

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Lesso

n 7

-7


7-7

MAPS Use the map shown and a

customary ruler to find the actual

distance between each pair of cities.

Measure to the nearest sixteenth of

an inch.

1. Port Jacob and Southport

2. Port Jacob and Brighton Beach

3. Brighton Beach and Pirates’ Cove

4. Eastport and Sand Dollar Reef

5. SCALE MODEL Sanjay is making a 139 centimeters long scale model of the Parthenon

for his World History class. The actual length of the Parthenon is 69.5 meters long.


b. How many times as long as the actual Parthenon is the model?

6. ARCHITECTURE An architect is making a scale model of an office building he wishes to

construct. The model is 9 inches tall. The actual office building he plans to construct will

be 75 feet tall.


b. What scale factor did the architect use to build his model?

7. WHITE HOUSE Craig is making a scale drawing of the White House on an

8.5-by-11-inch sheet of paper. The White House is 168 feet long and 152 feet wide.

Choose an appropriate scale for the drawing and use that scale to determine the

drawing’s dimensions.

8. GEOGRAPHY Choose an appropriate scale and construct a scale drawing of each

rectangular state to fit on a 3-by-5-inch index card.

a. The state of Colorado is approximately 380 miles long (east to west) and 280 miles

wide (north to south).

b. The state of Wyoming is approximately 365 miles long (east to west) and 265 miles

wide (north to south).

Skills Practice


Port Jacob

Eastport Brighton Beach

Pirates’ Cove

Southport

Sand Dollar Reef

0.5 in.

20 mi

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Practice


7-7

MAPS Use the map of Central New Jersey shown and an inch ruler to find the

actual distance between each pair of cities. Measure to the nearest sixteenth of

an inch.

1. Highland Park and Metuchen

2. New Brunswick and Robinvale

3. Rutgers University Livingston

Campus and Rutgers University

Cook–Douglass Campus

4. AIRPLANES William is building a

scale model of a Boeing 747–400 aircraft.

The wingspan of the model is approximately 8 feet 10 1−16

inches. If the scale factor of the

model is approximately 1:24, what is the actual wingspan of a Boeing 747–400 aircraft?

5. ENGINEERING A civil engineer is making a scale model of a highway on ramp.

The length of the model is 4 inches long. The actual length of the on ramp is 500 feet.


b. How many times as long as the actual on ramp is the model?

c. How many times as long as the model is the actual on ramp?

6. MOVIES A movie director is creating a scale model of the Empire State Building to

use in a scene. The Empire State Building is 1250 feet tall.

a. If the model is 75 inches tall, what is the scale of the model?

b. How tall would the model be if the director uses a scale factor of 1:75?

7. MONA LISA A visitor to the Louvre Museum in Paris wants to sketch a drawing

of the Mona Lisa, a famous painting. The original painting is 77 centimeters by

53 centimeters. Choose an appropriate scale for the replica so that it will fit on a

8.5-by-11-inch sheet of paper.

1 in.

1.88 mi

HighlandPark

Metuchen

NewBrunswick

Robinvale

RutgersUniv-Cook-Douglass

Campus

RutgersUniv-Livingston

Campus

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Lesso

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-7


7-7

1. MODELS Luke wants to make a scale

model of a Boeing 747 jetliner. He wants

every foot of his model to represent

50 feet. Complete the following table.

Part

Actual

length (in.)

Model

length (in.)

Wing Span 2537

Length 2782

Tail Height 392

(Source: Boeing)

2. PHOTOGRAPHS Tracy is 4 feet tall and

her father is 6 feet tall. In a photograph

of the two of them standing side by side,

Tracy’s image is 2 inches tall.

6 ft

4 ft

Although their images are much

smaller, the ratio of their heights

remains the same. How tall is Tracy’s

father’s image in the photo? What is

the scale of the photo?

3. TOWERS The Tokyo Tower in Japan is

currently the world’s tallest self-

supporting steel tower. It is 333 meters

tall.

a. Heero builds a model of the Tokyo

Tower that is 2775 millimeters tall.

What is the scale of Heero’s model?

b. How many times as tall as the actual

tower is the model?

4. PUPPIES Meredith’s new Pomeranian

puppy is 7 inches tall and 9 inches long.

She wants to make a drawing of her new

Pomeranian to put in her locker. If the

sheet of paper she is using is 3 inches by

5 inches, find an appropriate scale factor

for Meredith to use in her drawing.

5. MAPS Carlos makes a map of his

neighborhood for a presentation. The

scale of his map is 1 inch:125 feet.

a. How many feet do 4 inches represent

on the map?

b. Carlos lives 250 feet away from

Andrew. How many inches separate

Carlos’ home from Andrew’s on

the map?

c. During a practice run in front of his

parents, Carlos realizes that his map

is far too small. He decides to make

his map 5 times as large. What would

be the scale of the larger map?



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7-7

Area and Volume of Scale Models and Drawings

You have already learned about changes of length measures between scale models and the object that is being modeled. The areas and volumes of scale models and drawings also change, but by multiples different than the “scale factor.”

Yuan is making a scale model of a cylinder. The actual cylinder has

a radius of r inches and a height of h inches. The scale factor of the model is 1:2.

What is the ratio of volumes of the model cylinder to the actual cylinder?

The volume formula for a cylinder is V = πr2h. The actual cylinder’s volume is πr2h.

If the cylinder is scaled down by a factor of 1:2, the new radius will be 1 − 2 r and the new

height will be 1 − 2 h.

V = π ( 1 − 2 r)

2

( 1 − 2 h)

= 1 − 8 πr2h

The model’s volume is 1 − 8 of the actual volume.

Exercises

1. Consider a painting on a 22-inch-by-28-inch canvas. Suppose you wish to make a 1:2 scale model of the painting for art class. What is the ratio of areas of the model painting to the actual painting?

(Area = Length × Width)

2. The Parks and Recreation Office is planning a new circular playground with a radius of 30 feet. Before they can construct the playground, they ask an architect to create a 1:20 scale model of the proposed playground such that the new radius is 1.5 feet. What is the ratio of areas of the model playground to the proposed actual playground?

(Area = π × (radius)2)

3. A refrigerator manufacturer uses a 7-foot-by-3-foot-by-3-foot box for its standard model. The marketing team suggests the manufacturer start selling a smaller, lower-priced refrigerator with a scale factor of 4:5 to the standard model. If the box is reduced by a similar scale, what is the ratio of volumes of the new, smaller box to the current box?

(Volume = length × width × height)

4. Consider a cube with side lengths x. If each side of the cube is scaled by a factor of 1:y, what is the ratio of volumes of the model cube to the actual cube?

(Volume = length × width × height)

Enrichment

Example

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Assessment


8.

7

Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Multiple Choice

Student Recording Sheet

Use this recording sheet with pages 610–611 of the Student Edition.

Record your answer in the blank.

For gridded response questions, also enter your answer in the grid by writing

each number or symbol in a box. Then fill in the corresponding circle for that

number or symbol.

8. (grid in)

9.

10.

11.

12.

13.

14. (grid in)

15.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Extended Response

Record your answers for Question 16 on the back of this paper.

14.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Short Response/Gridded Response

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SCORE


7

General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show

how he or she arrived at the answer, the student will be awarded only 1 credit.

All extended-response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for

all parts of the question. For example, if a question has three parts, the correct

response to one or two parts of the question that required work to be shown is not

considered a fully correct response.

• Students who use trial and error to solve a problem must show their method.

Merely showing that the answer checks or is correct is not considered a complete

response for full credit.

Exercise 11 Rubric

Rubric for Extended-Response

Score Speci! c Criteria

4 An understanding that using proportional parts created by parallel lines is

shown by the student creating the proportion XQ

− QZ

= YR −

RZ . The correct

substitutions are made for part b to determine the length of −−

RZ , or 16 units.

The correct substitutions are made for part c to determine the length of −−

XY , or 9.5 units.

3 A generally correct solution, but may contain minor flaws in reasoning or computation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0 An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.

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7

SCORE

SCORE


7

1. The ratios of measures of the angles in △ABC is 4:13:19. Find

the measures of the angles.

2. SCHOOL ELECTIONS Henrietta conducted a random survey

of 60 students and found that 36 are planning to vote for her

as class president. If there are 460 students in Henrietta’s

class, predict the total number of students who will vote for

her as class president.

3. Are any of the three triangles

similar? If so, write

the appropriate similarity

statement.

4. If △ABC ∼ △DEC, find x and the

scale factor of △ABC to △DEC.

5. MULTIPLE CHOICE In a rectangle, the ratio of the length to

the width is 5:2, and its perimeter is 126 centimeters. Find the

width of the rectangle.

A 9 cm B 18 cm C 45 cm D 50.4 cm

D

E

CB

A

20

25

3x - 22x

BP X Y

Z

Q

R

A

C

5

4 8 8 8

10

4

1.

2.

3.

4.

5.

For Questions 1 and 2, determine whether each pair of

triangles is similar. Justify your answer.

1. 2.

3. Identify the similar triangles in the figure,

then find the value of x.

4. Determine whether −−−

MN ‖ −−

KL

Justify your answer.

5. In △ABC, −−−

DE is parallel to −−

AC and DE = 10. Find the length

of −−

AC if −−−

DE is the midsegment of △ABC .

1.

2.

3.

4.

5.

Chapter 7 Quiz 1(Lessons 7-1 and 7-2)


8

A

B D

E

F

C 4

6x

20

15

98

10 12

41°

52°41°

87°

J

M

N11

2

3

9

L

K

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7

SCORE

SCORE


7

For Questions 1–3 △ABC ∼ △DEF, and −−

BE bisects ∠ABC.

1. Find the length of −−

XC to the nearest tenth.

2. What is the relationship between the corresponding altitudes

−− BX and

−− EY ?


EY to the nearest tenth.

4. Determine if the dilation from A and B is an enlargement or reduction. Then find the scale factor of the dilation.

5. Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: A(-1, 4), B(1, 4), C(4, -2), D(-4, -2)

image: F(-3, 0), G (- 7 −

3 , 0) , H (- 4 −

3 , -2) , J(-4, -2)

1.

2.

3.

4.

5.

1. GARDENS The model of a circular garden is 8 inches in diameter. The actual garden will be 20 feet in diameter. Find the ratio of the diameter of the model to the diameter of the actual garden.

2. PHOTOS A 4-inch by 6-inch photograph, set vertically, is enlarged to make a poster 22 inches wide. How tall is the poster?

3. TICKETS Sofia is making a copy of a ticket from the school play to put in her memory album. The original ticket is 7 inches long and 5 inches wide. Her copy is 2 inches long.

What is the scale of the copy of the ticket?

4. MULTIPLE CHOICE An NCAA regulation size lacrosse field measures 110 yards long by 60 yards wide. Which of the following would be an appropriate scale to construct a scale drawing of a lacrosse field so it would best fit on a 8.5-by-11-inch sheet of paper? (1 yd = 36 in.)

A 1:250 B 1:255 C 1:355 D 1:365

1.

2.

3.

4.


Chapter 7 Quiz 4(Lesson 7-7)

12 14 cm

2 cm

8 cm

X

y

8

y

x

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7

1. Polygon ABCD is similar to polygon PQRS. Which proportion must be true?

A AC

− AD

= PQ

− PS

B BC

− CD

= QR

− RS

C AB −

BD =

PQ −

QR D

CD −

AB =

PQ −

RS

2. This fall, 126 students participated in the soccer program, while 54 played

volleyball. What was the ratio of soccer players to volleyball players?

F 3 −

4 G

3 −

7 H 4 −

3 J

7 −

3

3. The ratio of the measures of the angles of a triangle is 2:3:10. What is the

least angle measure?

A 12 B 15 C 24 D 36

4. Find the value of x.

F 2 H 6

G 4.8 J 6.4

5. Rectangle ABCD ∼ rectangle EFGH, the perimeter of ABCD is

54 centimeters and the perimeter of EFGH is 36 centimeters. What is the

scale factor of ABCD to EFGH?

A 2 − 3 B

3 −

2 C

3 −

5 D

5 −

3

1.

2.

3.

4.

5.

Part I Write the letter for the correct answer in the blank at the right of each question.

Chapter 7 Mid-Chapter Test(Lessons 7-1 through 7-4)

10 8

x

D

E

ACB18

6.

7.

8.

9.

10.

Part II

For Questions 6 and 7, determine whether each pair of

triangles is similar. Justify your answer.

6. 7.

8. Determine whether −−−

MN ‖ −−

KL . Justify your answer.

4525181085°85°6

415

10

J

M

N11

2

3

9

L

K

20

12

15DE

B

A

C

9. Find DE.

10. RECTANGLES A rectangle has a perimeter of 14 inches. A

similar rectangle has a perimeter of 10 inches. If the length of

the larger rectangle is 4 inches, what is the length of the

smaller rectangle? Round to the nearest tenth.

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7

Choose the correct term to complete each sentence.

1. If there are 15 girls and 9 boys in an art class, the (ratio, scale factor) of girls to boys in the class is 5:3.

2. If △ABC ∼ △DEF, AB = 10, and DE = 2.5, then the (scale factor, proportion) of △ABC to △DEF is 4:1.

Choose from the terms above to complete each sentence.

3. In △LMN, P lies on −−−

LM and Q lies on −−−

LN . If PQ = 1 − 2 MN,

−−− PQ

is called a(n) .

4. The product of the in the equation 3 − x = 24

− 30

is 90.

5. The product of the in the equation 3 − x = 24

− 30

is 24x.

Write whether each sentence is true or false. If false,

replace the underlined term to make a true sentence.

6. If quadrilaterals ABCD and WXYZ have corresponding angles congruent and corresponding sides proportional, they are called cross products.

7. The equation 3 − x = 24

− 30

is called a(n) scale factor.

8. In a proportion, the cross products of the extremes equals the cross product of the means.

Define each term in your own words.

9. equality of cross products

10. midsegment

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

cross products

dilation

enlargement

extremes

means


proportion

ratio

reduction

scale

scale drawing

scale factor

scale model

similar polygons

similarity transformation

Chapter 7 Vocabulary Test

?

?

?

Assessment

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7

Write the letter for the correct answer in the blank at the right of each question.

1. There are 15 plums and 9 apples in a fruit bowl. What is the ratio of apples

to plums?

A 3:5 B 3:8 C 5:3 D 8:3

2. The scale drawing of a porch is 8 inches wide by 12 inches long. If the actual

porch is 12 feet wide, what is the length of the porch?

F 8 ft G 10 ft H 16 ft J 18 ft

3. Solve 5 −

6 = 4 − x .

A 4.6 B 4.8 C 5 D 7

4. A quality control technician checked a sample of 30 bulbs. Two of the bulbs

were defective. If the sample was representative, find the number of bulbs

expected to be defective in a case of 450.

F 24 G 30 H 36 J 45

5. Find the triangle similar to △ABC at the right.

A C

B D

6. Find the value of x if △ABC ∼ △JKL.

F 10 H 25

G 14 J 29

7. Quadrilateral ABCD ∼ quadrilateral PQRS. If AB = 10, BC = 6, PS = 12,

and QR = 4, find the scale factor of ABCD to PQRS.

A 1 − 2 B

3 −

2 C

5 −

3 D

5 −

6

8. Quadrilateral ABCD ∼ quadrilateral EFGH.

Find the value of x.

F 15 H 25

G 20 J 30

9. Which theorem or postulate can be used to prove

that these two triangles are similar?

A AA B SAS C SSA D SSS

10. Find MN.

F 5 1 − 3 G 6

3 −

4 H 7 J 12

2210

24

15

36

39

6

84

27°63°

B

C

D

A

F

E

H

G60° 120°

80°100°4x°

A C

B J L

K18

6 9x - 2

4

35

5

12

3

5

12

13

CB

A

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Chapter 7 Test, Form 1

35°

8

96

35°

Q

PM

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7

11. A 5-foot tall student cast a 4-foot shadow. If the tree next to her cast a 44-foot

shadow, what is the height of the tree?

A 35 1 − 5 ft B 45 ft C 51 1 −

2 ft D 55 ft

12. In △ABC, −−−

DE ‖ −−

AC . If AD = 12, BD = 3, and CE = 10,

find BE.

F 1 H 2

G 1 1 − 2 J 2 1 −

2

13. What is the scale factor of the dilation of A to B?

A 1 C 2

B 3 −

2 D 6

14. △FGH ∼ △PQR, FG = 6, PQ = 10, and the perimeter

of △PQR is 35. What is the perimeter of △FGH?

F 21 G 27 H 31 J 58 1 − 3


A 14 C 16

B 15 D 18

16. △LMN ∼ △XYZ with altitudes −−

KL

and −−−

WX . Find KL.

F 6 H 9

G 7 J 19


A 5 C 6 1 − 2

B 6 D 7 1 − 2


F 16 H 20

G 18 J 21

19. Nathan is building a model of his father’s sailboat with a scale factor of 1 − 32

.

The actual sail is in the shape of a right triangle with a base of 8 meters and

a hypotenuse of 13 meters. What will be the approximate perimeter of the

sail on the model boat?

A 32 cm B 48.81 cm C 65.62 cm D 97.65 cm

Bonus In △ABC, AB = 10, BC = 16,

−−−

DE ⊥ −−

AC , and DE = 6. Find CD.

15

1824

x

16

10

129x

28 123

K

L X

M YWZN

16

10

24x

11.

12.

13.

14.

15.

16.

17.

18.

19.

Chapter 7 Test, Form 1 (continued)

C B

A

E

D

12

10

3

106

B C

D

E

A

16

B:

−2−1

y

x

2

1

−2

−11 2 3 4 5 6

Assessment

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7


1. Of the 240 students eating lunch, 96 purchased their lunch and the rest

brought a bag lunch. What is the ratio of students purchasing lunch to

students bringing a bag lunch?

A 2:3 B 2:5 C 3:2 D 5:2

2. In a rectangle, the ratio of the width to the length is 4:5. If the rectangle is

40 centimeters long, find its width.

F 32 cm G 36 cm H 44 cm J 50 cm

3. A postage stamp 25 millimeters wide and 40 millimeter tall is enlarged to

make a poster. The poster is 4 feet wide. Find the height of the poster.

A 2.5 ft B 5.25 ft C 5.8 ft D 6.4 ft

4. Find the polygon that is similar to ABCD.

F H

G J

5. If △PQR ∼ △STU, find the value of x.

A 4.4 C 24.6

B 7 D 35

6. If ABCD ∼ EFGH, find the value of x.

F 18.75 H 22.75

G 20 J 28

7. △ABC ∼ △LMN, AB = 18, BC = 12, LN = 9, and LM = 6. What is the scale

factor of △ABC to △LMN?

A 9 −

2 B

3 −

2 C

3 −

1 D 2 −

1

8. Name the theorem or postulate that can be used to

prove that these triangles are similar.

F AA Similarity H SAS Similarity

G SSS Similarity J SSA Similarity

For Questions 9 and 10, refer to the figure at the right.

9. Identify the true statement.

A △PQR ˜ △RST C △PQR ˜ △TSR

B △PQR ˜ △STR D △PQR ˜ △TRS


F 2 1 − 2 G 3 H 3 1 −

2 J 4

2 6

2–3

2

15

18

4x

2

21–2

3

T

R

P

Q

S

14

4

3

7

16

30

10

x - 4

A

BC

DE H

GF

22°123°

5x°12 8

4.5

R U

S

TQ

P

8

8

18

6 4

12

A D

B C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Chapter 7 Test, Form 2A

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7 Chapter 7 Test, Form 2A (continued)

11. A 24-foot flagpole cast a 20-foot shadow. At the same time, the building next

to it cast an 85-foot shadow. Find the height of the building.

A 70 5 −

6 ft B 89 ft C 96 1 −

6 ft D 102 ft

12. Find QT.

F 15 H 19

G 17 J 21

13. Find the value of x so that −−

ST ‖ −−

PR .

A 4 C 6

B 4 1 − 2 D 6 1 −

2

14. Find the value of y.

F 4 −

3 H

7 −

3

G 2 J 3

15. If △KLM ∼ △XYZ, find the perimeter of △XYZ.

A 40 C 45

B 42 D 48

16. △ABC ∼ △JKL with altitudes −−

BX and −−

KY .Find BX.

F 19.2 H 24.6

G 21 J 28


A 4 C 6

B 5 D 8


F 2 H 1

G 3 −

2 J

1 −

2

Bonus Find the value of x.

128

x + 2 x

32

XCA

B10

6

YLJ

K

9

X Z

Y4

K M

L7

9

6

9

2y

1–3y + 2

25

5

34x - 3QR

P

S

T

24

40

35RQ

PS

T

11.

12.

13.

14.

15.

16.

17.

18.

B:

12

4 x

−3−2−1

y

x

4

3

2

1

−4

−4

−3

−2

−11 2 3 4

Assessment

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NAME DATE PERIOD

SCORE


7 Chapter 7 Test, Form 2B


1. Given the choice between doing an oral and a written report, 18 of the 28

students chose to do an oral report. Find the ratio of written to oral reports.

A 5:9 B 9:5 C 9:14 D 14:9

2. A model of a lighthouse has diameter 8 inches and height 18 inches. If the

actual diameter of the lighthouse is 20 feet, find its actual height.

F 30 ft G 35 ft H 45 ft J 50 ft

3. The three sides of a triangle are in the ratio 2:4:5. If the shortest side of the

triangle is 4 meters long, find the perimeter.

A 17 m B 22 m C 32 m D 40 m

4. Find the polygon that is similar to ABCD.

F H

G J

5. If △ABC ∼ △DEF, find m∠C.

A 54 C 67

B 59 D 69

6. Quadrilateral JKLM ∼ quadrilateral WXYZ, JK = 15, LM = 10, XY = 6, and

WX = 9. Find KL.

F 8 G 10 H 11 J 12

7. △LMN ∼ △RST, LN = 21, MN = 28, and the scale factor of △RST to △LMN

is 4 − 3 . Find ST.

A 15 3 −

4 B 21 C 28 D 37 1 −

3

8. Name the theorem or postulate that can be used to prove that these triangles

are similar.

F AA Similarity H SSA Similarity

G SAS Similarity J SSS Similarity

For Questions 9 and 10, refer to the figure at the right.

9. Identify the similar triangles.

A △LMN ∼ △MPQ C △LMN ∼ △QPM

B △LMN ∼ △QMP D △LMN ∼ △PQM

10. Find LM.

F 16 G 17 H 18 J 20

22

18

6

17

14

6

1510 12

L

N

MP

Q

20

8

11.2

28

54°67°

67°

DECA

FB

12

15

5

12

15

54

A B

D C

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

30

24

8

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NAME DATE PERIOD


7

11.

12.

13.

14.

15.

16.

17.

18.

B:

11. A 6-foot-tall fence post cast a 2 1 − 2 -foot shadow. At the same time, a nearby

clock tower cast a 35-foot shadow. What is the height of the tower?

A 37 1 − 2 ft B 71 ft C 78 ft D 84 ft

12. Find CE.

F 25 H 27

G 26 J 28

13. Find FH so that −−−

GH ‖ −−−

DE .

A 4 C 6

B 5 D 7


F 4 H 9

G 6 J 12


A 1 −

2 C

4 −

3

B 3 −

4 D 2

16. △ABC ∼ △XYZ with altitudes −−

AP and

−−− XQ . Find AP.

F 11 1 − 4 H 15 1 −

4

G 14 J 20


A 7 C 9

B 8 D 10


F 3 3 −

4 H 6

G 4 J 7 1 − 2

Bonus Find CE.6

5

4

39

A

B

C

D

E

6

21–4

16

y

R T

SU

40

16

x + 2

3x

125

Q

X

YZ

27

P

A

B C

5

8

2x

x + 3

9

10

15D

G

FE H

8

20

EB C

DA

35

Chapter 7 Test, Form 2B (continued)

−3−2−1

y

x

4

3

2

1

−4−1

1 2 3 4

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NAME DATE PERIOD

SCORE 7

1. Of the 300 television sets sold at an electronics store last

month, 90 were flat-screen TVs. What is the ratio of

flat-screen TVs to other TVs sold last month?

2. Determine whether △ABC ∼ △DEF. Justify your answer.

22.5

15 1510

A

B

C DE

F

3. When a 5-foot vertical pole casts a 3-foot 4-inch shadow, an

oak tree casts a 20-foot shadow. Find the height of the tree.

4. Quadrilateral ABCD ∼ quadrilateral WXYZ, AB = 15,

BC = 27, BC = 27, and the scale factor of WXYZ to

ABCD is 2 − 3 . Find XY.

5. The blueprint for a swimming pool is 8 inches by 2 1 − 2 inches.

The actual pool is 136 feet long. Find the width of the pool.

6. Find CD.

1.8

2.46C E

G

F

D

7. If quadrilateral ABCD ∼ quadrilateral PQRS, find BC.

28

106.5

R

S P

Q B

A

C

D

8. Is the dilation a similarity

transformation? Verify your answer.

9. △ABC ∼ △XYZ, AB = 12, AC = 16, BC = 20, and XZ = 24.

Find the perimeter of △XYZ.

For Questions 10 and 11, use the figure.

10. Identify the similar triangles.

11. Find the value of x.15 2x - 4

x

6

P

Q

R S

T

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 7 Test, Form 2C


−6−4−2

y

x

8

6

4

2

−22 4 6 8 10

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7

12. If △ABC ∼ △PQR and −−−

BM and −−−

QN are medians, find BM.

11

4.43.8

A C P R

Q

B

M N

13. The ratio of the measures of the three sides of a triangle is

3:4:6. If the perimeter is 91, find the length of the longest

side.

14. If △RST ∼ △UVW, find m∠W.

15. In △ABC, −−−

AX bisects ∠BAC.


16. Find the value of y so that −−−

MN || −−−

BC .

17. △ABC ∼ △LMN, and −−−

AD and 36

208

D

A

CB M P

L

N

−−

LP are altitudes. Find AD.

18. Find the value of x. 8

18

2x - 2

13.5

Bonus Find EG. 18

10

12

B

F

D E

G

C

A

28

4x - 5

2x

8

6

A C

X

B

12.

13.

14.

15.

16.

17.

18.

B:

Chapter 7 Test, Form 2C (continued)


85°

48°R T

S

U W

V

18

8y4

y

B

CAN

M

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NAME DATE PERIOD

SCORE 7

1. Of the 112 students in the marching band, 35 were in the

drum section. What is the ratio of drummers to other

musicians in the band?

2. Determine whether quadrilateral ABCD ∼ quadrilateral

EFGH. Justify your answer.

A

9

7

14 17

9 8.5

13.5

12B

C

DH G

FE

3. When a 9-foot tall garden shed cast a 5-foot 3-inch shadow,

a house cast a 28-foot shadow. Find the height of the house.

4. △ABC ∼ △FGH, AB = 24, AC = 16, GH = 9, and FH = 12.

Find the scale factor of △ABC to △FGH.

5. The model of a suspension bridge is 18 inches long and

2 inches tall. If the length of the actual bridge is 1650 feet,

find its height.

6. Find GP.

2.73.3

4.5

F H

Q

G

P

7. If △JKL ∼ △PQR,

find the value of x.

8. Is the dilation a similarity

transformation? Verify your answer.

9. △ABC ∼ △PQR, AB = 18, BC = 20, AC = 22, and QR = 25.

Find the perimeter of △PQR.

For Questions 10 and 11, use the figure.

20

x

6

Y Z

X

M

N

16 10. Identify the similar triangles.

11. Find MN.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 7 Test, Form 2D


−6−4−2

y

x

8

6

4

2

−22 4 6 8 10

35

14

6

x

J L

PQ

R

K

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7

12. If △FGH ∼ △LMN and −−

AF and −−

BL are medians, find BL.

16.8 4.8

7.7

F H L

MB

N

G

A

13. The ratio of the measures of the three angles of a triangle is

3:4:8. Find the measure of the largest angle.

14. If quadrilateral DEFG ∼ quadrilateral WXYZ, find m∠Y.

E F Z W

X

YG

D

68° 85°

82°125°

15. In △PQR, −−−

QS bisects ∠PQR.


16. Find the value of x so that −−−

PQ || −−−

FH .

17. If △FGH ∼ △JKL, find GX.

X Y

K

F H JL

27 12 21–2

G


42 2y

12 16

Bonus Find FG.

12.

13.

14.

15.

16.

17.

18.

B:

Chapter 7 Test, Form 2D (continued)


P Q

R

S3x - 1

32

40

2x + 3

63

3x2x - 1

F H

Q

G

P

2.8 3.24.2

8 7

A

B F

GEC

D

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NAME DATE PERIOD

SCORE 7

1. In an orchard of apple and peach trees, 3 −

7 of the trees are

peaches. What is the ratio of apple trees to peach trees?

2. Determine whether trapezoid ABCD ∼ trapezoid PQRS.

Justify your answer.

9.6

15.6

12.47.8

9

15

6

23.4D

R S

PQ

C

A

B

3. In △ABC, m∠A = 51, AB = 14, and AC = 20. In △DEF,

m∠D = 51, DE = 16.8, and DF = 24. Determine whether

△ABC ∼ △DEF. Justify your answer.

4. A painting that is 48 inches by 12 inches is reduced to fit on a

canvas that is 30 centimeters by 10 centimeters. Find the

maximum dimensions of the reduced painting.

5. Find the value of x. 6

4.8

6.5K L

J

N

M

2x

6. If △ABC ∼ △XYZ, find y.

17.5A C

B

2y

14X Z

Y

y + 3

7. The ratio of the measures of the sides of a triangle is 2:5:6. If

the length of the longest side is 48 inches, find the perimeter.

8. Find m∠ I.

F

G

HI

J

45°

53°

9. △ABC ∼ △DEF, AB = 8, BC = 13, AC = 15, and DF = 20.

Find the perimeter of △DEF.

10. △ABC ∼ △JKL, AB = 12, BC = 18.4, KL = 6.9, and JL = 5.6.

Find the scale factor of △ABC to △JKL.

11. Find the value of y. y + 84y - 7

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 7 Test, Form 3


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NAME DATE PERIOD

7

12. Find SR.

30 12

Q

RPS

40

For Questions 13 and 14, △FGH ∼ △JKL.

12x

F H

G7.5

4.5

J L

K


14. Find the ratio of the perimeter of △FGH to the perimeter of △JKL.

15. Find AD so that −−−

DE || −−

AB .

A B

ED

C

2x - 4 3x

8x5x

16. A triangle with coordinates A(0, 0), B(4, 0), and C(0, 4) is enlarged by a factor of 2. What are the coordinates of the image?

17. When a 15-foot tall climbing wall cast a 20-foot shadow, a building cast a 32-foot shadow. Find the height of the building.

18. If △ABC ∼ △EFG and −−−

BD and −−−

FH are medians, find BD.

HE G

F2x - 5

75

1–4

x + 5

DA C

B

Bonus The ratio of the lengths of the

sides of △ABC is 5:2 1 − 2 :4. The

scale factor of △ABC to △DEC is 5:2, and the scale factor of △DEC to △FEG is 1:4.

−−− BC is

the shortest side and −−

AB is the longest side of △ABC. Find FG.

12.

13.

14.

15.

16.

17.

18.

B:

Chapter 7 Test, Form 3 (continued)


10

A

BC

D E

G

F

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NAME DATE PERIOD

SCORE 7

Demonstrate your knowledge by giving a clear, concise solution to

each problem. Be sure to include all relevant drawings and justify

your answers. You may show your solution in more than one way or

investigate beyond the requirements of the problem.

1. Write a possible proportion if the extremes are 3 and 10.

2. △ABC and △WXY are isosceles triangles.

a. Write a possible ratio for the sides of △ABC if its perimeter is

42 inches.

b. Name possible measures for the sides of △ABC using your answer to

part a.

c. If △WXY has a perimeter of 28 and △ABC has sides with the measures

you gave in part b, what must be the measure of the sides of △WXY so

that △WXY ∼ △ABC?

3. Write as many triangle similarity statements as possible for the figure

below. How do you know that these triangles are similar?

A B C D

IE

F

G

H

4. Sketch two triangles that are not similar, but have one pair of

corresponding angles congruent and two pairs of corresponding sides

proportional. Label the corresponding angles and the proportional sides.

5. Draw △XYZ inside △PQR with half the perimeter of △PQR. Explain your

process and why it works.

Chapter 7 Extended-Response Test


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NAME DATE PERIOD

SCORE 7


YZ . (Lesson 1-2) X Z

5.5 in.

3.6 in. Y

A 1.9 in. C 7.2 in. B 5.3 in. D 12.5 in.

2. Given: 3b + 4 < 16 Conjecture: b > 0Which of the following would be a counterexample? (Lesson 2-1)

F b = -1 G b = 0 H b = 3.5 J b = 4

3. Find the plane that is parallel to

R

P

Q

ST

U

plane PTU. (Lesson 3-1)

A plane QRU C plane PQS B plane QRS D plane SPU

4. Find m∠1. (Lesson 3-2)

(3x + 5)°

(125 - 7x)°

1 F 5 H 41 G 12 J 44

5. Which statement must be true in order to D

A C

B

prove △ABC % △DCB by SAS? (Lesson 4-4)

A −−−

CB bisects ∠ABD. B ∠BCA % ∠CBD C ∠BDC % ∠CAB D

−− AB %

−−− BC

6. In an indirect proof, you assume that the conclusion is false and then find a(n) ? . (Lesson 5-4)

F assumption H truth value G contradiction J conditional statement

7. Demont and Tony are competing to see whose house is the tallest. Early in the afternoon, Tony, who is 4 feet tall, measured his shadow to be 9.6 inches and the shadow of his house to be 62.4 inches. Later in the day, Demont, who is 5 feet tall, measured his shadow to be 15.6 inches and the shadow of his house to be 62.4 inches. Who lives in the taller house? (Lesson 7-3)

A Demont B Both houses are the same height. C Tony D There is not enough information.

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.

1.

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Standardized Test Practice(Chapters 1-7)


A B C D

Assessment

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NAME DATE PERIOD

7

8. Find the coordinates of the midpoint of −−

AB for A(-24, 15) and B(13, -31). (Lesson 1-3)

F (-18.5, -23) G (-11, -16) H (-5.5, -8) J (10.5, 23)

For Questions 9 and 10, use the figure D E

G F

at the right.

9. The perimeter of rectangle DEFG is 176, EF = h, and DE = 7h. What is the value of h? (Lesson 1-6)

A 11 C 22 B 15 D 77

10. What is the area of the rectangle DEFG? (Lesson 1-6)

F 88 units2 G 225 units2 H 513 units2 J 847 units2

11. What is the slope-intercept form for the line y + 7 = 4(x-10)? (Lesson 3-4)

A y = 4x - 47 B 4x - y = 47 C 4x = y + 17 D 4 x − y = 17

12. Which of the following is the equation of a line parallel to the line passing through (4, -3) and (8, 5)? (Lesson 3-4)

F y = x + 2 G 2y = 9x + 4 H 2y = 2x + 4 J y = 2x + 9

8. F G H J

9. A B C D

10. F G H J

11. A B C D

12. F G H J

13. Find m∠2. (Lesson 3-2)

52°

12

34

14. △LMN is equilateral, LM is one more than three times a number, MN is nine less than five times the number, and NL is eleven more than the number. Find LM. (Lesson 4-1)

Part 2: Gridded Response

Instructions: Enter your answer by writing each digit of the

answer in a column box and then shading in the appropriate circle

that corresponds to that entry.

13.

14.

Standardized Test Practice (continued)


9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

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7

Part 3: Short Response

Instructions: Write your answers in the spaces provided.

15. Solve -16

− 40

= 4x + 10

− 5 . (Lesson 7-1)

16. −−

XA is an altitude of △XYB. Find the value of y. (Lesson 5-2)

17. Two sides of a triangle measure 21 inches and 32 inches, and the third side measures x inches. Find the range for the value of x. (Lesson 5-5)

18. If △ABC ∼ △HIJ, find the perimeter of △HIJ. (Lesson 7-2)

19. Given T(3, -1), U(1, -7), V(8, -5), W(2, 6), X(-4, 8), and Y(-2, 1), determine whether △TUV $ △WXY. Explain. (Lesson 4-4)

20. Two parallel lines are cut by a transversal, ∠1 and ∠2 are adjacent angles, m∠1 = 12y + 10, and m∠2 = 20y - 34. Find m∠1 and m∠2. (Lesson 3-2)

21. Use points S(-5, 7), T(1, 9), P(12, -1), and R(3, 26).

a. Find the lengths of −−

ST and −−

PR to the nearest hundredth. (Lesson 1-3)

b. Determine the slope of −−

ST and of −−

PR . (Lesson 3-3)

c. Are −−

ST and −−

PR parallel, perpendicular, or neither? (Lesson 3-3)

32 cm

80 cm

60 cm

A B

C12 cm

H I

J

(2y - 81)°B

A

X

Y

15.

16.

17.

18.

19.

20

21a.

21b.

21c.

Standardized Test Practice (continued)


Answers

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

NA

ME

DA

TE

PE

RIO

D

Lesson 7-1

Ch

ap

ter

7

5

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dy

Gu

ide a

nd

In

terv

en

tio

n

Rati

os a

nd

Pro

po

rtio

ns

Wri

te a

nd

Use

Ra

tio

s A

ra

tio

is

a c

om

pari

son

of

two q

uan

titi

es

by d

ivis

ion

s. T

he

rati

o a

to b

, w

here

b i

s n

ot

zero

, ca

n b

e w

ritt

en

as

a

−

b o

r a

:b.

In

20

07

th

e B

osto

n R

ed

So

x b

ase

ba

ll t

ea

m w

on

96

ga

me

s o

ut

of

16

2

ga

me

s p

lay

ed

. W

rit

e a

ra

tio

fo

r t

he

nu

mb

er o

f g

am

es w

on

to

th

e t

ota

l n

um

be

r o

f

ga

me

s p

lay

ed

.

To f

ind

th

e r

ati

o,

div

ide t

he n

um

ber

of

gam

es

won

by t

he t

ota

l n

um

ber

of

gam

es

pla

yed

. T

he

resu

lt i

s 9

6

−

162 ,

wh

ich

is

abou

t 0.5

9.

Th

e B

ost

on

Red

Sox w

on

abou

t 59%

of

their

gam

es

in

2007.

Th

e r

ati

o o

f th

e m

ea

su

re

s o

f th

e a

ng

les

in △

JH

K i

s 2

:3:4

. F

ind

th

e m

ea

su

re

s o

f th

e a

ng

les.

Th

e e

xte

nd

ed

rati

o 2

:3:4

can

be r

ew

ritt

en

2x:3

x:4

x.

Sk

etc

h a

nd

label

the a

ngle

measu

res

of

the t

rian

gle

. T

hen

wri

te a

nd

solv

e a

n e

qu

ati

on

to f

ind

th

e v

alu

e o

f x.

2x +

3x +

4x =

180

Triangle

Sum

Theore

m

9x =

180

Com

bin

e lik

e t

erm

s.

x =

20

Div

ide e

ach s

ide b

y 9

.

Th

e m

easu

res

of

the a

ngle

s are

2(2

0)

or

40,

3(2

0)

or

60,

an

d 4

(20)

or

80.

Exerc

ises

1

. In

th

e 2

007 M

ajo

r L

eagu

e B

ase

ball

seaso

n,

Ale

x R

od

rigu

ez h

it 5

4 h

om

e r

un

s an

d w

as

at

bat

583 t

imes.

Wh

at

is t

he r

ati

o o

f h

om

e r

un

s to

th

e n

um

ber

of

tim

es

he w

as

at

bat?

5

4

−

58

3 o

r ab

ou

t 0.0

926

2. T

here

are

182 g

irls

in

th

e s

op

hom

ore

cla

ss o

f 305 s

tud

en

ts.

Wh

at

is t

he r

ati

o o

f gir

ls t

o

tota

l st

ud

en

ts?

1

82

−

30

5

3. T

he l

en

gth

of

a r

ect

an

gle

is

8 i

nch

es

an

d i

ts w

idth

is

5 i

nch

es.

Wh

at

is t

he r

ati

o o

f le

ngth

to w

idth

?

8

−

5

4. T

he

rati

o of

th

e si

des

of

a t

rian

gle

is

8:1

5:1

7. It

s peri

mete

r is

480 i

nch

es. F

ind t

he l

en

gth

of

each

sid

e of

th

e tr

ian

gle

.

96 i

n., 1

80 i

n., 2

04 i

n.

5. T

he

rati

o of

th

e m

easu

res

of t

he t

hre

e a

ngle

s of

a t

rian

gle

is

7:9

:20. F

ind t

he m

easu

re o

f each

an

gle

of

the

tria

ngle

.

35, 45, 100

Exam

ple

1

Exam

ple

2

7-1

4x°

2x°

3x°

Chapter Resources


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

3

Gle

nc

oe

Ge

om

etr

y

An

tici

pati

on

Gu

ide

Pro

po

rtio

ns a

nd

Sim

ilari

ty

7

B

efo

re y

ou

beg

in C

ha

pte

r 7

•

R

ead

each

sta

tem

en

t.

•

D

eci

de w

heth

er

you

Agre

e (

A)

or

Dis

agre

e (

D)

wit

h t

he s

tate

men

t.

•

W

rite

A o

r D

in

th

e f

irst

colu

mn

OR

if

you

are

not

sure

wh

eth

er

you

agre

e o

r d

isagre

e,

wri

te N

S (

Not

Su

re).

A

fter y

ou

co

mp

lete

Ch

ap

ter 7

•

R

ere

ad

each

sta

tem

en

t an

d c

om

ple

te t

he l

ast

colu

mn

by e

nte

rin

g a

n A

or

a D

.

•

D

id a

ny o

f you

r op

inio

ns

abou

t th

e s

tate

men

ts c

han

ge f

rom

th

e f

irst

colu

mn

?

•

F

or

those

sta

tem

en

ts t

hat

you

mark

wit

h a

D,

use

a p

iece

of

pap

er

to w

rite

an

exam

ple

of

wh

y y

ou

dis

agre

e.

Ste

p 2

Ste

p 1

ST

EP

1A

, D

, o

r N

SS

tate

men

tS

TE

P 2

A o

r D

1

. R

ati

os

are

alw

ays

wri

tten

as

fract

ion

s.

2

. A

pro

port

ion

is

an

equ

ati

on

sta

tin

g t

hat

two r

ati

os

are

equ

al.

3

. T

wo r

ati

os

are

in

pro

port

ion

to e

ach

oth

er

on

ly i

f th

eir

cr

oss

pro

du

cts

are

equ

al.

4

. If

a

−

b =

c

−

d , t

hen

ad

= b

c.

5

. T

he r

ati

o o

f th

e l

en

gth

s of

the s

ides

of

sim

ilar

figu

res

is

call

ed

th

e s

cale

fact

or

for

the t

wo f

igu

res.

6

. If

on

e a

ngle

in

a t

rian

gle

is

con

gru

en

t to

an

an

gle

in

an

oth

er

tria

ngle

, th

en

th

e t

wo t

rian

gle

s are

sim

ilar.

7

. If

a l

ine i

s p

ara

llel

to o

ne s

ide o

f a t

rian

gle

an

d i

nte

rsect

s th

e o

ther

two s

ides

in t

wo d

isti

nct

poin

ts,

then

th

e l

ine

sep

ara

tes

the t

wo s

ides

into

con

gru

en

t se

gm

en

ts.

8

. A

segm

en

t w

hose

en

dp

oin

ts a

re t

he m

idp

oin

ts o

f tw

o

sid

es

of

a t

rian

gle

is

para

llel

to t

he t

hir

d s

ide o

f th

e

tria

ngle

.

9

. If

tw

o t

rian

gle

s are

sim

ilar

then

th

eir

peri

mete

rs a

re

equ

al.

10

. T

he m

ed

ian

s of

two s

imil

ar

tria

ngle

s are

in

th

e s

am

e

pro

port

ion

as

corr

esp

on

din

g s

ides.

D A A D A D D A D A

Answers (Anticipation Guide and Lesson 7-1)

Chapter 7 A1 Glencoe Geometry

Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.

Answers (Lesson 7-1)



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

6

Gle

nc

oe

Ge

om

etr

y

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Rati

os a

nd

Pro

po

rtio

ns

Use

Pro

pe

rtie

s o

f P

rop

ort

ion

s A

sta

tem

ent

that

two

rati

os a

re

equ

al

is c

all

ed a

pro

po

rti

on

. In

th

e p

rop

orti

on a

−

b =

c −

d ,

wh

ere

b a

nd

d a

re

not

zer

o, t

he

valu

es a

an

d d

are

th

e e

xtr

em

es a

nd

th

e valu

es b

an

d c

are

th

e m

ea

ns.

In a

pro

por

tion

, th

e p

rod

uct

of

the

mea

ns

is e

qu

al

to t

he

pro

du

ct o

f th

e ex

trem

es,

so a

d =

bc.

Th

is i

s th

e C

ross

Pro

du

ct P

rop

erty

.

S

olv

e

9

−

16 =

27

−

x .

9

−

16 =

27

−

x

9 ·

x =

16

· 2

7

Cro

ss P

roducts

Pro

pert

y

9x =

432

Multip

ly.

x =

48

D

ivid

e e

ach s

ide b

y 9

.

PO

LIT

ICS

Ma

yo

r H

ern

an

de

z c

on

du

cte

d a

ra

nd

om

su

rv

ey

of

20

0 v

ote

rs a

nd

fo

un

d t

ha

t 1

35

ap

pro

ve

of

the

jo

b s

he

is d

oin

g.

If t

he

re

are

48

,00

0 v

ote

rs i

n M

ay

or H

ern

an

de

z’s

to

wn

, p

re

dic

t th

e t

ota

l n

um

be

r o

f v

ote

rs

wh

o a

pp

ro

ve

of

the

jo

b s

he

is d

oin

g.

Wri

te a

nd

sol

ve

a p

rop

orti

on t

hat

com

pare

s th

e n

um

ber

of

regis

tere

d v

oter

s an

d t

he

nu

mber

of

reg

iste

red

vot

ers

wh

o ap

pro

ve

of t

he

job t

he

mayor

is

doi

ng.

1

35

−

200 =

x

−

48,0

00

← v

ote

rs w

ho a

ppro

ve

← a

ll vote

rs

135 ·

48,0

00 =

20

0 ·

x

Cro

ss P

roducts

Pro

pert

y

6,4

80,0

00 =

20

0x

Sim

plif

y.

32,4

00 =

x

Div

ide e

ach s

ide b

y 2

00.

Base

d o

n t

he

surv

ey,

abou

t 32,4

00 r

egis

tere

d v

oter

s ap

pro

ve

of t

he

job t

he

mayor

is

doi

ng.

Exerc

ises

So

lve

ea

ch

pro

po

rti

on

.

1.

1

−

2 =

28

−

x

56

2.

3

−

8 =

y

−

24 9

3.

x +

22

−

x +

2

= 3

0

−

10 8

4.

3

−

18.2

= 9

−

y 54.6

5. 2

x +

3

−

8

=

5

−

4 3.5

6. x

+ 1

−

x -

1

= 3

−

4

-7

7. If

3 D

VD

s co

st $

44.8

5,

fin

d t

he

cost

of

one

DV

D.

$14.9

5

8

. B

OT

AN

Y B

ryon

is

mea

suri

ng p

lan

ts i

n a

fie

ld f

or a

sci

ence

pro

ject

. O

f th

e fi

rst

25

pla

nts

he

mea

sure

s, 1

5 o

f th

em a

re s

mall

er t

han

a f

oot

in h

eigh

t. I

f th

ere

are

4000

pla

nts

in

th

e fi

eld

, p

red

ict

the

tota

l n

um

ber

of

pla

nts

sm

all

er t

han

a f

oot

in h

eigh

t.

2

400 p

lan

ts

7-1

Exam

ple

1

Exam

ple

2

a

−

b =

c −

d

a ·

d =

b ·

c

extr

em

es

means

↑↑


NA

ME

DA

TE

PE

RIO

D

Lesson 7-1

Ch

ap

ter

7

7

Gle

nc

oe

Ge

om

etr

y

Sk

ills

Pra

ctic

e

Rati

os a

nd

Pro

po

rtio

ns

7-1

1. FO

OT

BA

LL A

tig

ht

end

sco

red

6 t

ouch

dow

ns

in 1

4 g

am

es.

Fin

d t

he

rati

o of

tou

chd

own

s p

er g

am

e.

0.4

3:1

2. E

DU

CA

TIO

N I

n a

sch

edu

le o

f 6 c

lass

es,

Mart

a h

as

2 e

lect

ive

class

es.

Wh

at

is t

he

rati

o of

ele

ctiv

e to

non

-ele

ctiv

e cl

ass

es i

n M

art

a’s

sch

edu

le?

1:2

3. B

IOLO

GY

Ou

t of

274 l

iste

d s

pec

ies

of b

ird

s in

th

e U

nit

ed S

tate

s, 7

8 s

pec

ies

mad

e th

e en

dan

ger

ed l

ist.

Fin

d t

he

rati

o of

en

dan

ger

ed s

pec

ies

of b

ird

s to

lis

ted

sp

ecie

s in

th

e U

nit

ed S

tate

s.

39:1

37

4. B

OA

RD

GA

ME

S M

yra

is

pla

yin

g a

boa

rd g

am

e. A

fter

12 t

urn

s, M

yra

has

lan

ded

on

a

blu

e sp

ace

3 t

imes

. If

th

e gam

e w

ill

last

for

100 t

urn

s, p

red

ict

how

man

y t

imes

Myra

w

ill

lan

d o

n a

blu

e sp

ace

.

25

5. S

CH

OO

L T

he

rati

o of

male

stu

den

ts t

o fe

male

stu

den

ts i

n t

he

dra

ma c

lub a

t C

am

pbel

l H

igh

Sch

ool

is 3

:4.

If t

he

nu

mber

of

male

stu

den

ts i

n t

he

clu

b i

s 18,

pre

dic

t th

e n

um

ber

of

fem

ale

stu

den

ts?

24

So

lve

ea

ch

pro

po

rti

on

.

6

. 2

−

5 =

x

−

40

16

7

.

7

−

10 =

21

−

x

30

8

. 2

0

−

5

= 4

x

−

6

6

9

. 5

x

−

4

= 3

5

−

8

3.5

1

0. x

+1

−

3

= 7

−

2

9.5

1

1.

15

−

3

= x

-3

−

5

28

12

. Th

e ra

tio

of t

he

mea

sure

s of

th

e si

des

of

a t

rian

gle

is

3:5

:7,

an

d i

ts p

erim

eter

is

450 c

enti

met

ers.

Fin

d t

he

mea

sure

s of

each

sid

e of

th

e tr

ian

gle

.

90 c

m,

150 c

m,

210 c

m

13

. Th

e ra

tio

of t

he

mea

sure

s of

th

e si

des

of

a t

rian

gle

is

5:6

:9, an

d i

ts p

erim

eter

is

220 m

eter

s.

Wh

at

are

th

e m

easu

res

of t

he

sid

es o

f th

e tr

ian

gle

?

55 m

, 66 m

, 99 m

14

. Th

e ra

tio

of t

he

mea

sure

s of

th

e si

des

of

a t

rian

gle

is

4:6

:8,

an

d i

ts p

erim

eter

is

126 f

eet.

W

hat

are

th

e m

easu

res

of t

he

sid

es o

f th

e tr

ian

gle

?

28 f

t, 4

2 f

t, 5

6 f

t

15

. Th

e ra

tio

of t

he

mea

sure

s of

th

e si

des

of

a t

rian

gle

is

5:7

:8,

an

d i

ts p

erim

eter

is

40 i

nch

es.

Fin

d t

he

mea

sure

s of

each

sid

e of

th

e tr

ian

gle

.

10 i

n., 1

4 i

n., 1

6 i

n.

Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.




NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

8

Gle

nc

oe

Ge

om

etr

y

Practi

ce

Rati

os a

nd

Pro

po

rtio

ns

1. N

UT

RIT

ION

O

ne o

un

ce o

f ch

ed

dar

cheese

con

tain

s 9 g

ram

s of

fat.

Six

of

the g

ram

s of

fat

are

satu

rate

d f

ats

. F

ind

th

e r

ati

o o

f sa

tura

ted

fats

to t

ota

l fa

t in

an

ou

nce

of

cheese

.

2

. FA

RM

ING

T

he r

ati

o o

f goats

to s

heep

at

a u

niv

ers

ity r

ese

arc

h f

arm

is

4:7

. T

he n

um

ber

of

sheep

at

the f

arm

is

28.

Wh

at

is t

he n

um

ber

of

goats

?

3

. Q

UA

LIT

Y C

ON

TR

OL

A w

ork

er

at

an

au

tom

obil

e a

ssem

bly

pla

nt

check

s n

ew

cars

for

defe

cts.

Of

the f

irst

280 c

ars

he c

heck

s, 4

have d

efe

cts.

If

10,5

00 c

ars

wil

l be c

heck

ed

th

is

mon

th,

pre

dic

t th

e t

ota

l n

um

ber

of

cars

th

at

wil

l h

ave d

efe

cts.

150 c

ars

So

lve

ea

ch

pro

po

rti

on

.

4.

5

−

8 =

x

−

12

5.

x

−

1.1

2 =

1

−

5

6.

6x

−

27 =

43

7.

x +

2

−

3

= 8

−

9

2

−

3

8.

3x -

5

−

4

= -

5

−

7

5

−

7

9.

x -

2

−

4

= x

+ 4

−

2

10. T

he r

ati

o o

f th

e m

easu

res

of

the s

ides

of

a t

rian

gle

is

3:4

:6,

an

d i

ts p

eri

mete

r is

104 f

eet.

Fin

d t

he m

easu

re o

f each

sid

e o

f th

e t

rian

gle

.

11. T

he r

ati

o o

f th

e m

easu

res

of

the s

ides

of

a t

rian

gle

is

7:9

:12,

an

d i

ts p

eri

mete

r is

84 i

nch

es.

Fin

d t

he m

easu

re o

f each

sid

e o

f th

e t

rian

gle

.

12. T

he r

ati

o o

f th

e m

easu

res

of

the s

ides

of

a t

rian

gle

is

6:7

:9,

an

d i

ts p

eri

mete

r is

77 c

en

tim

ete

rs.

Fin

d t

he m

easu

re o

f each

sid

e o

f th

e t

rian

gle

.

13. T

he r

ati

o o

f th

e m

easu

res

of

the t

hre

e a

ngle

s is

4:5

:6.

Fin

d t

he m

easu

re o

f each

an

gle

of

the t

rian

gle

.

14. T

he r

ati

o o

f th

e m

easu

res

of

the t

hre

e a

ngle

s is

5:7

:8.

Fin

d t

he m

easu

re o

f each

an

gle

of

the t

rian

gle

.

15. B

RID

GE

S A

con

stru

ctio

n w

ork

er

is p

laci

ng r

ivets

in

a n

ew

bri

dge. H

e u

ses

42 r

ivets

to b

uil

d

the f

irst

2 f

eet

of

the b

ridge. If

th

e b

ridge i

s to

be 2

200 f

eet

in l

en

gth

, pre

dic

t th

e n

um

ber

of

rivets

th

at

wil

l be n

eeded f

or

the e

nti

re b

ridge.

4

6,2

00 r

ivets

7-1 2

:3

16

7.5

0.2

24

193.5 -

10

24 f

t, 3

2 f

t, 4

8 f

t

21 i

n., 2

7 i

n., 3

6 i

n.

21 c

m,

24.5

cm

, 31.5

cm

48,

60,

72

45,

63,

72

Lesson 7-1


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

9

Gle

nc

oe

Ge

om

etr

y

Wo

rd

Pro

ble

m P

racti

ce

Rati

os a

nd

Pro

po

rtio

ns

1

. T

RIA

NG

LE

ST

he r

ati

os

of

the m

easu

res

of

the a

ngle

s in

△D

EF

is 7

:13:1

6.

13

x°

16

x° 7

x°

Fin

d t

he m

easu

re o

f th

e a

ngle

s.

3

5,65,80

2. R

AT

ION

S S

ixte

en

stu

den

ts w

en

t on

a w

eek

-lon

g h

ikin

g t

rip

. T

hey b

rou

gh

t

wit

h t

hem

320 s

peci

all

y b

ak

ed

, p

rote

in-

rich

, co

ok

ies.

Wh

at

is t

he r

ati

o o

f co

ok

ies

to s

tud

en

ts?

3. C

LO

VE

RS

N

ath

an

iel

is s

earc

hin

g f

or

a f

ou

r-le

af

clover

in a

fie

ld.

He f

ind

s

2 f

ou

r-le

af

clovers

du

rin

g t

he f

irst

12 m

inu

tes

of

his

searc

h.

If N

ath

an

iel

spen

ds

a t

ota

l of

180 m

inu

tes

searc

hin

g

in t

he f

ield

, p

red

ict

the n

um

ber

of

fou

r-

leaf

clovers

Nath

an

iel

wil

l fi

nd

.

30 c

lovers

4.C

AR

S A

car

com

pan

y b

uil

ds

two

vers

ion

s of

on

e o

f it

s m

od

els

—a s

ed

an

an

d a

sta

tion

wagon

. T

he r

ati

o o

f se

dan

s

to s

tati

on

wagon

s is

11:2

. A

fre

igh

ter

begin

s u

nlo

ad

ing t

he c

ars

at

a d

ock

.

Tom

cou

nts

18 s

tati

on

wagon

s an

d t

hen

overh

ears

a d

ock

work

er

call

ou

t, “

Ok

ay,

that’

s all

of

the w

agon

s . . .

bri

ng o

ut

the s

ed

an

s!”

How

man

y s

ed

an

s w

ere

on

the s

hip

?

99

5. D

ISA

ST

ER

RE

AD

INE

SS

Th

e t

ow

n o

f

Oyst

er

Bay i

s co

nd

uct

ing a

su

rvey o

f

80 h

ou

seh

old

s to

see h

ow

pre

pare

d i

ts

citi

zen

s are

for

a n

atu

ral

dis

ast

er.

Of

those

hou

seh

old

s su

rveyed

, 66 h

ave a

surv

ival

kit

at

hom

e.

a.

Wri

te t

he r

ati

o o

f p

eop

le w

ith

surv

ival

kit

s in

th

e s

urv

ey.

33

−

40 o

r 0.8

25

b.

Wri

te t

he r

ati

o o

f p

eop

le w

ith

ou

t

surv

ival

kit

s in

th

e s

urv

ey.

7

−

40 o

r 0.1

75

c.

Th

ere

are

29,0

00 h

ou

seh

old

s in

Oyst

er

Bay.

If t

he t

ow

n w

ish

es

to

pu

rch

ase

su

rviv

al

kit

s fo

r all

hou

seh

old

s th

at

do n

ot

curr

en

tly h

ave

on

e,

pre

dic

t th

e n

um

ber

of

kit

s it

wil

l

have t

o p

urc

hase

.

ab

ou

t 5075 k

its

7-1 2

0:1

Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

10

G

len

co

e G

eo

me

try

En

rich

men

t

Gro

wth

Ch

art

sIt

is

said

th

at

wh

en

a c

hil

d h

as

reach

ed

th

e a

ge o

f 2 y

ears

, h

e i

s ro

ugh

ly h

alf

of

his

ad

ult

heig

ht.

Th

e g

row

th c

hart

belo

w s

how

s th

e g

row

th a

ccord

ing t

o p

erc

en

tile

s fo

r boys.

1

. U

se t

he c

hart

to d

ete

rmin

e t

he a

pp

roxim

ate

heig

ht

for

a b

oy a

t age 2

if

he i

s in

th

e 7

5th

to 9

5th

perc

en

tile

.

3

5–37 i

nch

es

2

. U

sin

g t

he r

ule

th

at

the h

eig

ht

at

age 2

is

ap

pro

xim

ate

ly h

alf

of

his

ad

ult

heig

ht,

set

up

a

pro

port

ion

to s

olv

e f

or

the a

du

lt h

eig

ht

of

the b

oy i

n E

xerc

ise 1

. S

olv

e y

ou

r p

rop

ort

ion

.

36

−

x

= 1

−

2 ;

72 i

nch

es

3

. U

se t

he c

hart

to a

pp

roxim

ate

th

e h

eig

ht

at

age 1

8 f

or

a b

oy i

f h

e i

s in

th

e 7

5th

to 9

5th

perc

en

tile

. H

ow

does

this

an

swer

com

pare

to t

he a

nsw

er

to p

roble

m 1

?

7

2–74 i

nch

es:

the t

wo

an

sw

ers

are

very

clo

se

4

. R

ep

eat

this

pro

cess

for

a b

oy w

ho i

s in

th

e 5

th t

o 2

5th

perc

en

tile

.

3

2–34 i

nch

es;

66 i

nch

es;

65–68 i

nch

es

5

. Is

usi

ng t

he r

ule

th

at

a b

oy i

s h

alf

of

his

ad

ult

heig

ht

at

age 2

years

a g

ood

ap

pro

xim

ati

on

? E

xp

lain

.

Y

es,

it a

pp

ears

to

be a

go

od

ap

pro

xim

ati

on

.

7-1

85

75

65

55

45

35

24

68

10

12

14

16

18

Ag

e (

yr)

Height (in.)

Perc

en

tile

Gro

up

5th

to

25th

25th

to

75th

75th

to

95th

A

B


NA

ME

DA

TE

PE

RIO

D

Lesson 7-1

Ch

ap

ter

7

11

G

len

co

e G

eo

me

try

You

can

use

a c

alc

ula

tor

to s

olv

e p

rop

ort

ion

s.

S

olv

e t

he

pro

po

rti

on

by

usin

g c

ro

ss p

ro

du

cts

. R

ou

nd

yo

ur a

nsw

er

to t

he

ne

are

st

hu

nd

re

dth

.

55.6

−

16.9

= 4

5.8

−

x

Mu

ltip

ly 1

6.9

by 4

5.8

. D

ivid

e t

he p

rod

uct

by 5

5.6

.

En

ter:

16.9

3

45.8

÷

55.6

E

NT

ER

13.92122302

Th

e s

olu

tion

is

ap

pro

xim

ate

ly 1

3.9

2

So

lve

ea

ch

pro

po

rti

on

by

usin

g c

ro

ss p

ro

du

cts

. R

ou

nd

yo

ur a

nsw

ers

to t

he

ne

are

st

hu

nd

re

dth

.

1

. 1

3.9

−

39.8

=

10.5

−

x

2

. 2

5.9

−

37.7

=

24.3

−

x

3

. 1

9.6

−

54.3

=

27.7

−

x

4

. 6

6.8

−

43.4

=

x

−

16.9

5

. 7

5.4

−

37.2

=

x

−

32.4

6

.

x

−

46.2

=

29.7

−

36.4

7

.

x

−

14.9

=

16.8

−

24.6

8

. 3

5.8

−

x

=

32.9

−

27.8

9

. 4

6.9

−

x

=

15.7

−

99.9

10

. 3

4.9

−

21.1

=

x

−

36.6

1

1.

x

−

99.8

=

32.2

−

45.3

1

2. 6

8.9

−

x

=

44.3

−

86.4

x ≈

30.0

6

x ≈

35.3

7

x ≈

76.7

4

7-1

Gra

ph

ing C

alc

ula

tor

Act

ivit

y

So

lvin

g P

rop

ort

ion

s

Exam

ple

x ≈

26.0

1

x ≈

65.6

7

x ≈

37.7

0

x ≈

10.1

8

x ≈

30.2

5

x ≈

298.4

3

x ≈

60.5

4

x ≈

70.9

4

x ≈

134.3

8



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

12

G

len

co

e G

eo

me

try

Ide

nti

fy S

imil

ar

Po

lyg

on

s S

imil

ar

poly

gon

s h

ave t

he s

am

e s

hap

e b

ut

not

nece

ssari

ly

the s

am

e s

ize.

If △

AB

C ∼

△X

YZ

, li

st

all

pa

irs o

f co

ng

ru

en

t a

ng

les a

nd

writ

e a

pro

po

rti

on

th

at

re

late

s t

he

co

rre

sp

on

din

g s

ide

s.

Use

th

e s

imil

ari

ty s

tate

men

t.

Con

gru

en

t an

gle

s: ∠

A !

∠X

, ∠

B !

∠Y

, ∠

C !

∠Z

Pro

port

ion

: A

B

−

XY

= B

C

−

YZ

= C

A

−

ZX

De

term

ine

wh

eth

er t

he

pa

ir o

f fi

gu

re

s i

s s

imil

ar.

If s

o,

writ

e t

he

sim

ila

rit

y s

tate

me

nt

an

d s

ca

le f

acto

r.

Ex

pla

in y

ou

r r

ea

so

nin

g.

Ste

p 1

C

om

pare

corr

esp

on

din

g a

ngle

s.∠

W !

∠P

, ∠

X !

∠Q

, ∠

Y !

∠R

, ∠

Z !

∠S

Corr

esp

on

din

g a

ngle

s are

con

gru

en

t.

Ste

p 2

C

om

pare

corr

esp

on

din

g s

ides.

WX

−

PQ

=

12

−

8

= 3

−

2 ,

XY

−

QR

= 1

8

−

12 =

3

−

2 ,

YZ

−

RS

= 1

5

−

10 =

3

−

2 ,

an

d

Z

W

−

SP

=

9

−

6 =

3

−

2 .

Sin

ce c

orr

esp

on

din

g s

ides

are

pro

port

ion

al,

WX

YZ

∼ P

QR

S.

Th

e p

oly

gon

s are

sim

ilar

wit

h a

sca

le f

act

or

of

3

−

2 .

Exerc

ises

Lis

t a

ll p

air

s o

f co

ng

ru

en

t a

ng

les,

an

d w

rit

e a

pro

po

rti

on

th

at

re

late

s t

he

co

rre

sp

on

din

g s

ide

s

for e

ach

pa

ir o

f sim

ila

r p

oly

go

ns.

1

. △

DE

F ∼

△G

HJ

∠D

# ∠

G,

∠E

# ∠

J,

∠F #

∠H

; D

E

−

GJ =

EF

−

HJ =

FD

−

HG

2

. P

QR

S ∼

TU

WX

∠P

# ∠

T,

∠Q

# ∠

U,

∠R

# ∠

W,

∠S

# ∠

X;

PQ

−

TU

=

QR

−

UW

= R

S

−

WX =

SP

−

XT

De

term

ine

wh

eth

er e

ach

pa

ir o

f fi

gu

re

s i

s s

imil

ar.

If s

o,

writ

e t

he

sim

ila

rit

y

sta

tem

en

t a

nd

sca

le f

acto

r.

If n

ot,

ex

pla

in y

ou

r r

ea

so

nin

g.

3

.

4.

22

11

7-2

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Sim

ilar

Po

lyg

on

s

Exam

ple

1

15

9

12

18

XW

ZY

10

6

8

12

QP S

R

Exam

ple

2

no

, ∠

L ≇

∠E

△P

QR

∼ △

PS

T;

scale

facto

r 1

−

2

Lesson 7-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

13

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Sim

ilar

Po

lyg

on

s

Use

Sim

ila

r Fig

ure

s Y

ou

can

use

sca

le f

act

ors

an

d p

rop

ort

ion

s to

fin

d m

issi

ng s

ide

len

gth

s in

sim

ilar

poly

gon

s.

Th

e t

wo

po

lyg

on

s a

re

sim

ila

r.

Fin

d x

an

d y

.

x

y38

32

16

13

T

S

RP

N

M

Use

th

e c

on

gru

en

t an

gle

s to

wri

te t

he

corr

esp

on

din

g v

ert

ices

in o

rder.

△R

ST

∼ △

MN

P

Wri

te p

rop

ort

ion

s to

fin

d x

an

d y

.

3

2

−

16 =

x

−

13

38

−

y

= 3

2

−

16

16x =

32(1

3)

32y =

38(1

6)

x =

26

y =

19

If

△D

EF

∼ △

GH

J,

fin

d t

he

sca

le f

acto

r o

f △

DE

F t

o △

GH

J a

nd

th

e

pe

rim

ete

r o

f e

ach

tria

ng

le.

Th

e s

cale

fact

or

is

E

F

−

HJ =

8

−

12 =

2

−

3 .

Th

e p

eri

mete

r of

△D

EF

is

10 +

8 +

12 o

r 30.

2

−

3 =

Peri

mete

r of

△D

EF

−

−

P

eri

mete

r of

△G

HJ

Theore

m 7

.1

2

−

3 =

30

−

x

Substitu

tion

(3)(

30)

= 2

x

Cro

ss P

roducts

Pro

pert

y

45

= x

S

olv

e.

So,

the p

eri

mete

r of

△G

HJ

is

45.

Exerc

ises

Ea

ch

pa

ir o

f p

oly

go

ns i

s s

imil

ar.

Fin

d t

he

va

lue

of

x.

1.

9

12

8 10

x

2.

10

4.5

x

2.5 5

3.

12

x+

1

18

24

18

4.

40

36

x+

15

30

18

5. If

AB

CD

∼ P

QR

S, fi

nd t

he s

cale

fact

or

of A

BC

D t

o P

QR

S a

nd t

he

peri

mete

r

of e

ach

pol

ygon

.

s

cale

facto

r 4

−

5 ;

peri

mete

r A

BC

D =

68;

p

eri

mete

r P

QR

S =

85;

7-2

Exam

ple

1Exam

ple

2

12

12

10

8

14

20

25

x

= 6

x

= 2

.25

x =

15

x =

9



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

14

G

len

co

e G

eo

me

try

De

term

ine

wh

eth

er e

ach

pa

ir o

f fi

gu

re

s i

s s

imil

ar.

If s

o,

writ

e t

he

sim

ila

rit

y

sta

tem

en

t a

nd

sca

le f

acto

r.

If n

ot,

ex

pla

in y

ou

r r

ea

so

nin

g.

1

.

2.

7.5

7.5

7.5

7.5

ZY

XW

3

3

33

SR

QP

Ea

ch

pa

ir o

f p

oly

go

ns i

s s

imil

ar.

Fin

d t

he

va

lue

of x

.

3

.

4

.

5

.

6

.

9

10 3

x+

1

5

P

S

R

UV

WT

Q

10

8

x+

2

x-

1

SP

MN

L

T

x+

5

x+

5

9

3

44

S

W

XY

T U26

14

12

13

BC

DA

13

7

6x G

EH

F

10.5

69

59°

35°

AC

B

7

46

59°

35°

DF

E

△A

BC

∼ △

DE

F;

∠A

# ∠

D,

rh

om

bu

s P

QR

S ∼

rh

om

bu

s W

XY

Z;

∠C

# ∠

F,

an

d ∠

B #

∠E

∠P

# ∠

W,

∠Q

# ∠

X,

∠R

# ∠

Y,

by t

he T

hir

d A

ng

le T

heo

rem

, an

d

∠

S #

∠Z

; P

Q

−

WX

=

QR

−

XY =

R

S

−

YZ

=

SP

−

ZW

;

A

B

−

DE

=

BC

−

EF =

C

A

−

FD

; scale

facto

r: 3

−

2 .

scale

facto

r:

3

−

7.5

=

0.4

6

.5

7

5

13

Sk

ills

Practi

ce

Sim

ilar

Po

lyg

on

s

7-2

Lesson 7-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

15

G

len

co

e G

eo

me

try

Practi

ce

Sim

ilar

Po

lyg

on

sD

ete

rm

ine

wh

eth

er e

ach

pa

ir o

f fi

gu

re

s i

s s

imil

ar.

If s

o,

writ

e t

he

sim

ila

rit

y

sta

tem

en

t a

nd

sca

le f

acto

r.

If n

ot,

ex

pla

in y

ou

r r

ea

so

nin

g.

1.

25

12

9

14

.4

15

24

20

15

JM

QRS

PK

L

2.

24

12

14

16

21

18

VU

AC

BT

Ea

ch

pa

ir o

f p

oly

go

ns i

s s

imil

ar.

Fin

d t

he

va

lue

of x.

3.

14

10

x+

6

x+

9

CN

MD

AB

LP

4.

61

2

40°

40°

x+

1

x-

3A

F

E

D

BC

5

. P

EN

TA

GO

NS

If

AB

CD

E ∼

PQ

RS

T,

fin

d t

he

scale

fact

or

of

AB

CD

E t

o P

QR

ST

an

d t

he

peri

mete

r of

each

poly

gon

.

6. S

WIM

MIN

G P

OO

LS

T

he M

inn

itte

fam

ily

an

d t

he n

eig

hbori

ng G

au

det

fam

ily b

oth

have i

n-g

rou

nd

sw

imm

ing p

ools

. T

he

Min

nit

te f

am

ily p

ool,

PQ

RS

, m

easu

res

48 f

eet

by 8

4 f

eet.

Th

e G

au

det

fam

ily p

ool,

WX

YZ

, m

easu

res

40 f

eet

by 7

0 f

eet.

Are

the t

wo p

ools

sim

ilar?

If

so,

wri

te t

he

sim

ilari

ty s

tate

men

t an

d s

cale

fact

or.

7-2

20

15

10

21

48 ft

84 ft

70 ft

40 ft

P Q

SW Z

X Y

R

3

−

2

7

J

KLM

∼ Q

RS

P;

△

AB

C ∼

△U

VT;

∠A

# ∠

U,

∠J #

∠Q

, ∠

K #

∠R

,

∠

B #

∠V

, an

d ∠

C #

∠T

by t

he

∠L

# ∠

S,

∠M

# ∠

P,

an

d

Th

ird

An

gle

Th

eo

rem

an

d

JK

−

QR

= K

L

−

RS =

LM

−

SP

= M

J

−

PQ

= 5

−

3 o

r 1.6

7

A

B

−

UV =

BC

−

VT

= C

A

−

TU

= 2

−

3 .

scale

facto

r 5

−

7 ;

peri

mete

r A

BC

DE

= 6

5;

peri

mete

r P

QR

ST =

91

yes;

PQ

RS

∼ W

XY

Z;

scale

facto

r 6

−

5



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

16

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Sim

ilar

Po

lyg

on

s

1

. P

AN

ELS

W

hen

clo

sed

, an

en

tert

ain

men

t ce

nte

r is

mad

e o

f fo

ur

squ

are

pan

els

. T

he t

hre

e s

mall

er

pan

els

are

con

gru

en

t sq

uare

s.

Wh

at

is t

he s

cale

fact

or

of

the l

arg

er

squ

are

to o

ne o

f th

e s

mall

er

squ

are

s?

2

. W

IDE

SC

RE

EN

TE

LE

VIS

ION

S A

n

ele

ctro

nic

s co

mp

an

y m

an

ufa

ctu

res

wid

esc

reen

tele

vis

ion

sets

in

severa

l d

iffe

ren

t si

zes.

Th

e r

ect

an

gu

lar

vie

win

g

are

a o

f each

tele

vis

ion

siz

e i

s si

mil

ar

to

the v

iew

ing a

reas

of

the o

ther

sizes.

Th

e

com

pan

y’s

42-i

nch

wid

esc

reen

tele

vis

ion

h

as

a v

iew

ing a

rea p

eri

mete

r of

ap

pro

xim

ate

ly 1

44.4

in

ches.

Wh

at

is t

he

vie

win

g a

rea p

eri

mete

r of

the c

om

pan

y’s

46-i

nch

wid

esc

reen

tele

vis

ion

?

3

. IC

E H

OC

KE

Y A

n o

ffic

ial

Oly

mp

ic-s

ized

ic

e h

ock

ey r

ink

measu

res

30 m

ete

rs b

y

60 m

ete

rs.

Th

e i

ce h

ock

ey r

ink

at

the

loca

l co

mm

un

ity c

oll

ege m

easu

res

25.5

m

ete

rs b

y 5

1 m

ete

rs.

Are

th

e i

ce h

ock

ey

rin

ks

sim

ilar?

Exp

lain

you

r re

aso

nin

g.

4

. E

NLA

RG

ING

C

am

ille

wan

ts t

o m

ak

e

a p

att

ern

for

a f

ou

r-p

oin

ted

sta

r w

ith

d

imen

sion

s tw

ice a

s lo

ng a

s th

e o

ne

show

n.

Help

her

by d

raw

ing a

sta

r w

ith

dim

en

sion

s tw

ice a

s lo

ng o

n t

he

gri

d b

elo

w.

5

. B

IOLO

GY

A

para

meci

um

is

a s

mall

si

ngle

-cell

org

an

ism

. T

he p

ara

meci

um

m

agn

ifie

d b

elo

w i

s act

uall

y o

ne t

en

th

of

a m

illi

mete

r lo

ng.

a.

If y

ou

wan

t to

mak

e a

ph

oto

gra

ph

of

the o

rigin

al

para

meci

um

so t

hat

its

image i

s 1 c

en

tim

ete

r lo

ng,

by w

hat

scale

fact

or

shou

ld y

ou

magn

ify i

t?

b.

If y

ou

wan

t to

mak

e a

ph

oto

gra

ph

of

the o

rigin

al

para

meci

um

so t

hat

its

image i

s 15 c

en

tim

ete

rs l

on

g,

by w

hat

scale

fact

or

shou

ld y

ou

magn

ify i

t?

c.

By a

pp

roxim

ate

ly w

hat

scale

fact

or

has

the p

ara

meci

um

been

en

larg

ed

to

mak

e t

he i

mage s

how

n?

7-2 3 a

bo

ut

158.2

in

ch

es

100

1500

400

Yes;

the r

ati

o o

f th

e l

on

ger

dim

en

sio

ns o

f th

e r

inks i

s 2

0

−

17

an

d t

he r

ati

o o

f th

e s

maller

dim

en

sio

ns o

f th

e r

inks i

s 2

0

−

17 .

Lesson 7-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

17

G

len

co

e G

eo

me

try

En

ric

hm

en

t

Co

nstr

ucti

ng

Sim

ilar

Po

lyg

on

sH

ere

are

fou

r st

ep

s fo

r co

nst

ruct

ing a

poly

gon

th

at

is s

imil

ar

to a

nd

wit

h s

ides

twic

e a

s lo

ng a

s th

ose

of

an

exis

tin

g p

oly

gon

.

Ste

p 1

C

hoose

an

y p

oin

t eit

her

insi

de o

r ou

tsid

e t

he p

oly

gon

an

d l

abel

it O

.

Ste

p 2

D

raw

rays

from

O t

hro

ugh

each

vert

ex o

f th

e p

oly

gon

.

Ste

p 3

F

or

vert

ex V

, se

t th

e c

om

pass

to l

en

gth

OV

. T

hen

loca

te a

new

poin

t V

′ on

ray OV

su

ch t

hat VV

′ =

OV

. T

hu

s, OV

′ =

2(OV

).

Ste

p 4

R

ep

eat

Ste

p 3

for

each

vert

ex.

Con

nect

poin

ts V

′, W

′, X

′ an

d Y

′ to

fo

rm t

he n

ew

poly

gon

.

Tw

o c

on

stru

ctio

ns

of

poly

gon

s si

mil

ar

to a

nd

wit

h s

ides

twic

e t

hose

of VWXY

are

sh

ow

n

belo

w.

Noti

ce t

hat

the p

lace

men

t of

poin

t O

does

not

aff

ect

th

e s

ize o

r sh

ap

e o

f V

′W′X

′Y′,

on

ly i

ts l

oca

tion

.

VW

YX

VW

YX

V'

W'

X'

Y'

O

VW

YX

V'

W'

X'

Y'

O

Tra

ce

ea

ch

po

lyg

on

. T

he

n c

on

str

uct

a s

imil

ar p

oly

go

n w

ith

sid

es t

wic

e a

s l

on

g a

s

tho

se

of

the

giv

en

po

lyg

on

.

1

.

C

A

B

2

. D

EF

G

3

. E

xp

lain

how

to c

on

stru

ct a

sim

ilar

poly

gon

wit

h s

ides

thre

e t

imes

the

len

gth

of

those

of

poly

gon

HIJKL

. T

hen

d

o t

he c

on

stru

ctio

n.

4

. E

xp

lain

how

to c

on

stru

ct a

sim

ilar

poly

gon

1 1

−

2 t

imes

the l

en

gth

of

those

of

poly

gon

MNPQRS

. T

hen

do t

he

con

stru

ctio

n.

7-2

M

S

R

N

Q

P

HI

L

K

J

See s

tud

en

ts’

co

nstr

ucti

on

s.

See s

tud

en

ts’

wo

rk.

See s

tud

en

ts’

wo

rk.



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

18

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Sim

ilar

Tri

an

gle

s

Iden

tify

Sim

ilar

Tri

an

gle

s H

ere

are

th

ree w

ays

to s

how

th

at

two t

rian

gle

s are

sim

ilar.

AA

Sim

ila

rity

Tw

o a

ng

les o

f o

ne

tria

ng

le a

re c

on

gru

en

t to

tw

o a

ng

les o

f a

no

the

r tr

ian

gle

.

SS

S S

imil

ari

tyT

he

me

asu

res o

f th

e c

orr

esp

on

din

g s

ide

le

ng

ths o

f tw

o t

ria

ng

les a

re p

rop

ort

ion

al.

SA

S S

imil

ari

tyT

he

me

asu

res o

f tw

o s

ide

le

ng

ths o

f o

ne

tria

ng

le a

re p

rop

ort

ion

al to

th

e m

ea

su

res o

f tw

o

co

rre

sp

on

din

g s

ide

le

ng

ths o

f a

no

the

r tr

ian

gle

, a

nd

th

e in

clu

de

d a

ng

les a

re c

on

gru

en

t.

D

ete

rm

ine

wh

eth

er t

he

tria

ng

les a

re

sim

ila

r.

15

12

9

DE

F

10

86

AB

C

AC

−

DF

= 6

−

9 =

2

−

3

BC

−

EF

=

8

−

12 =

2

−

3

AB

−

DE

= 1

0

−

15 =

2

−

3

△A

BC

∼ △

DE

F b

y S

SS

Sim

ilari

ty.

D

ete

rm

ine

wh

eth

er t

he

tria

ng

les a

re

sim

ila

r.

8

4

70

°Q

RS

6

3

70

°

M

NP

3

−

4 =

6

−

8 ,

so M

N

−

QR

=

NP

−

RS

.

m∠

N =

m∠

R,

so ∠

N &

∠R

.

△N

MP

∼ △

RQ

S b

y S

AS

Sim

ilari

ty.

Exerc

ises

De

term

ine

wh

eth

er t

he

tria

ng

les a

re

sim

ila

r.

If s

o,

writ

e a

sim

ila

rit

y s

tate

me

nt.

Ex

pla

in y

ou

r r

ea

so

nin

g.

1

.

A

B

C

D

E

F

2

.

36

20

18

24

J

K L

WX

Y

3

. R

S

UL

T

4

.

18

8 65

°9

465

°

D

KJ

P

ZR

5

.

39

26

16

24

F

BE

L

MN

6

. 32

20

15

40

25

24G

H

QR

SI

7-3

Exam

ple

1Exam

ple

2

y

es;

△A

BC

∼ △

DE

F;

AA

Sim

ilari

ty

no

; 3

6

−

24 ≠

20

−

18

y

es;

△R

TU

∼ △

STL;

AA

Sim

ilari

ty

yes;

△K

DJ

∼ △

RP

Z;

SA

S S

imilari

ty

y

es;

△B

FE

∼ △

NLM

; S

AS

Sim

ilari

ty

yes;

△IG

H ∼

△S

QR

; S

SS

Sim

ilari

ty

Lesson 7-3


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

19

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Sim

ilar

Tri

an

gle

s

Use

Sim

ila

r T

ria

ng

les

Sim

ilar

tria

ngle

s ca

n b

e u

sed

to f

ind

measu

rem

en

ts.

△ABC

∼ △

DEF

. F

ind

th

e

va

lue

s o

f x a

nd

y.

18

√3

B CA

18

18

9

y

x

E FD

A

C

−

DF

= B

C

−

EF

A

B

−

DE

= B

C

−

EF

1

8 √

)

3

−

x

= 1

8

−

9

y

−

18 =

18

−

9

18x =

9(1

8 √

)

3 )

9y =

324

x =

9 √

)

3

y =

36

A

pe

rso

n 6

fe

et

tall

ca

sts

a 1

.5-f

oo

t-lo

ng

sh

ad

ow

at

the

sa

me

tim

e

tha

t a

fla

gp

ole

ca

sts

a 7

-fo

ot-

lon

g

sh

ad

ow

. H

ow

ta

ll i

s t

he

fla

gp

ole

?

7 f

t

6 f

t

?

1.5

ft

Th

e s

un

’s r

ays

form

sim

ilar

tria

ngle

s.

Usi

ng x

for

the h

eig

ht

of

the p

ole

, 6

−

x =

1.5

−

7 ,

so 1

.5x =

42 a

nd

x =

28

.

Th

e f

lagp

ole

is

28 f

eet

tall

.

Exerc

ises

ALG

EB

RA

Id

en

tify

th

e s

imil

ar t

ria

ng

les.

Th

en

fin

d e

ach

me

asu

re

.

1

. J

L

2. IU

3

. Q

R

4. B

C

5

. L

M

6. Q

P

7

. T

he h

eig

hts

of

two v

ert

ical

post

s are

2 m

ete

rs a

nd

0.4

5 m

ete

r. W

hen

th

e s

hort

er

post

ca

sts

a s

had

ow

th

at

is 0

.85 m

ete

r lo

ng,

wh

at

is t

he l

en

gth

of

the l

on

ger

post

’s s

had

ow

to

th

e n

eare

st h

un

dre

dth

?

x3

2

10

22

30

PQ

N

R

F

x

9

87.2

16

K

VP

M

L

24

36

√2

x

36

36

QV

R

S

T

35

13

10

20

JLK

Y

XZ

x

x

60

38

.6

23

30

E

BA

CD

20 13

26

I UW

G

H

x

7-3

Exam

ple

1Exam

ple

2

△

XY

Z ˜

△JK

L;

26

△

GIU

˜ △

GH

W;

30

△

SQ

V ˜

△S

RT;

12

△

BE

D ˜

△B

CA

; 19.3

△

KV

M ˜

△LP

M;

18

△

FR

Q ˜

△FN

P;

10 2

−

3

3.7

8 m



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.

NA

ME

DA

TE

PE

RIO

D


Ch

ap

ter

7

20

G

len

co

e G

eo

me

try

De

term

ine

wh

eth

er e

ach

pa

ir o

f tr

ian

gle

s i

s s

imil

ar.

If s

o,

writ

e a

sim

ila

rit

y

sta

tem

en

t. I

f n

ot,

wh

at

wo

uld

be

su

ffic

ien

t to

pro

ve

th

e t

ria

ng

les s

imil

ar? E

xp

lain

yo

ur r

ea

so

nin

g.

1

.

2.

3

.

4.

21

70°

15

MJ

P

14

70°

10

U

S

T

30°

60°

RQM

S

K

T

ALG

EB

RA

Id

en

tify

th

e s

imil

ar t

ria

ng

les.

Th

en

fin

d e

ach

me

asu

re

.

5. AC

6. JL

15

12

D

EC

BAx+

1

x+

5

1

6

4

M

NL

J

K

x+

18

x-

3

7. EH

8. VT

12

9

6

9

D

E

FH G

x+

5

6

14

S

U

V

RT

3x-

3

x+

2

12

12

8

9

96

CP

QR

A

B

Y

es;

△R

ST ∼

△W

SX

(o

r △

XS

W)

Yes;

△A

BC

∼ △

PQ

R (

or

S

AS

Sim

ilari

ty

△Q

PR

) b

y S

SS

Sim

ilari

ty

Y

es;

△S

TU

∼ △

JP

M b

y

Y

es;

△S

KM

∼ △

RTQ

by

S

AS

Sim

ilari

ty

A

A S

imilari

ty

△

AB

C ∼

△D

BE

; 16

△

JK

L ∼

△M

NL;

28

△

DE

F ∼

△G

EH

; 9

△

RS

T ∼

△U

VT;

5.4

Sk

ills

Practi

ce

Sim

ilar

Tri

an

gle

s

7-3

Lesson 7-3


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

21

G

len

co

e G

eo

me

try

Practi

ce

Sim

ilar

Tri

an

gle

s

De

term

ine

wh

eth

er t

he

tria

ng

les a

re

sim

ila

r.

If s

o,

writ

e a

sim

ila

rit

y s

tate

me

nt.

If

no

t, w

ha

t w

ou

ld b

e s

uff

icie

nt

to p

ro

ve

th

e t

ria

ng

les s

imil

ar? E

xp

lain

yo

ur

re

aso

nin

g.

1

.

2.

ALG

EB

RA

Id

en

tify

th

e s

imil

ar t

ria

ng

les.

Th

en

fin

d e

ach

me

asu

re

.

3. LM

, QP

4. NL

, ML

5

.

6.

8

N

JL

K

Mx+

5

6x+

2

24

12

18

N

P

QL

M

x+

3

x-

1

42°

42°

24

18

12

16

J

AK

YS

W

7-3

10

5

12

8

86

x +

7x -

1

x +

1

x +

33

4

7

. IN

DIR

EC

T M

EA

SU

RE

ME

NT

A

lig

hth

ou

se c

ast

s a 1

28-f

oot

shad

ow

. A

nearb

y l

am

pp

ost

that

measu

res

5 f

eet

3 i

nch

es

cast

s an

8-f

oot

shad

ow

.

a.

Wri

te a

pro

port

ion

th

at

can

be u

sed

to d

ete

rmin

e t

he h

eig

ht

of

the l

igh

thou

se.

b.

Wh

at

is t

he h

eig

ht

of

the l

igh

thou

se?

yes;

△JA

K ∼

△W

SY

; S

AS

Sim

ilari

ty

△LM

N ∼

△P

QN

; M

L =

12;

QP

= 8

△

JLN

∼ △

KLM

; LN

= 2

1;

LM

= 1

4

△TP

S ∼

△Q

PR

; P

S =

12;

PR

= 1

6

△E

GF ∼

△H

GI;

EG

= 6

; H

G =

8

Sam

ple

an

sw

er:

If

x =

heig

ht

of

lig

hth

ou

se,

x

−

5.2

5 =

12

8

−

8

.

84 f

tNo

. T

he t

rian

gle

s w

ou

ld b

e s

imilar

by S

AS

or

AA

. If

−−

MN

‖ −

−

PQ

.



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

22

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Sim

ilar

Tri

an

gle

s A

B

D

E

C

1

. C

HA

IRS

A

loca

l fu

rnit

ure

sto

re s

ell

s tw

o

vers

ion

s of

the s

am

e c

hair

: on

e f

or

ad

ult

s, a

nd

on

e f

or

chil

dre

n.

Fin

d t

he

valu

e o

f x s

uch

th

at

the c

hair

s are

sim

ilar.

18˝

16˝

X

12

˝

2

. B

OA

TIN

G T

he t

wo s

ail

boats

sh

ow

n a

re

part

icip

ati

ng i

n a

regatt

a.

Fin

d t

he

valu

e o

f x.

168.5

in.

220 in. X

in.

264 in.

40°

40°

3

. G

EO

ME

TR

Y G

eorg

ia d

raw

s a r

egu

lar

pen

tagon

an

d s

tart

s co

nn

ect

ing i

ts

vert

ices

to m

ak

e a

5-p

oin

ted

sta

r. A

fter

dra

win

g t

hre

e o

f th

e

lin

es

in t

he s

tar,

sh

e

beco

mes

curi

ou

s

abou

t tw

o t

rian

gle

s

that

ap

pear

in t

he

figu

re, △ABC

an

d

△CEB

. T

hey l

ook

sim

ilar

to h

er.

Pro

ve

that

this

is

the c

ase

.

4. S

HA

DO

WS

A

rad

io t

ow

er

cast

s a

shad

ow

8 f

eet

lon

g a

t th

e s

am

e t

ime

that

a v

ert

ical

yard

stic

k c

ast

s a s

had

ow

half

an

in

ch l

on

g.

How

tall

is

the r

ad

io

tow

er?

5. M

OU

NT

AIN

PE

AK

S G

avin

an

d B

rian

na

wan

t to

kn

ow

how

far

a m

ou

nta

in p

eak

is f

rom

th

eir

hou

ses.

Th

ey m

easu

re t

he

an

gle

s betw

een

th

e l

ine o

f si

te t

o t

he

peak

an

d t

o e

ach

oth

er’

s h

ou

ses

an

d

care

full

y m

ak

e t

he d

raw

ing s

how

n.

Gavin

Bri

anna

Peak

2 in.

2.0

15 in.

0.2

46 in.

90˚

83˚

Th

e a

ctu

al

dis

tan

ce b

etw

een

Gavin

an

d

Bri

an

na’s

hou

ses

is 1

1 − 2 m

iles.

a.

Wh

at

is t

he a

ctu

al

dis

tan

ce o

f th

e

mou

nta

in p

eak

fro

m G

avin

’s h

ou

se?

Rou

nd

you

r an

swer

to t

he n

eare

st

ten

th o

f a m

ile.

b.

Wh

at

is t

he a

ctu

al

dis

tan

ce o

f th

e

mou

nta

in p

eak

fro

m B

rian

na’s

hou

se?

Rou

nd

you

r an

swer

to t

he n

eare

st

ten

th o

f a m

ile.

7-3

13.5

in

.

202.2

in

.

S

am

ple

an

sw

er:

m∠

AD

B =

108,

so

m∠

DB

A =

36 (

base o

f is

o-

scele

s △

AB

D).

Th

us,

m∠

AB

C =

72.

Sim

ilarl

y,

m∠

DC

B =

36 a

nd

m∠

AC

B =

72.

So

, m∠

BE

C =

72

an

d m∠

BA

C =

36.

Th

ere

fore

,

△A

BC

an

d △

BC

E a

re s

imilar.

576 f

t

12.2

mi

12.3

mi

Lesson 7-3


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

23

G

len

co

e G

eo

me

try

Mo

vin

g S

had

ow

s

Have y

ou

ever

watc

hed

you

r sh

ad

ow

as

you

walk

ed

alo

ng t

he s

treet

at

nig

ht

an

d o

bse

rved

how

its

sh

ap

e c

han

ges

as

you

move?

Su

pp

ose

a m

an

wh

o i

s 6 f

eet

tall

is

stan

din

g b

elo

w

a l

am

pp

ost

th

at

is 2

0 f

eet

tall

. T

he m

an

is

walk

ing a

way f

rom

th

e l

am

pp

ost

at

a r

ate

of

5 f

eet

per

seco

nd

.

1

. If

th

e m

an

is

movin

g a

t a r

ate

of

5 f

eet

per

seco

nd

, m

ak

e a

con

ject

ure

as

to t

he r

ate

that

his

sh

ad

ow

is

movin

g.

2

. H

ow

far

aw

ay f

rom

th

e l

am

pp

ost

is

the m

an

aft

er

8 s

eco

nd

s?

3

. H

ow

far

is t

he e

nd

of

his

sh

ad

ow

fro

m t

he b

ott

om

of

the l

am

pp

ost

aft

er

8 s

eco

nd

s? U

se s

imil

ar

tria

ngle

s to

solv

e t

his

pro

ble

m.

4

. A

fter

3 m

ore

seco

nd

s, h

ow

far

from

th

e l

am

pp

ost

is

the m

an

? H

ow

far

from

th

e l

am

pp

ost

is h

is s

had

ow

?

5

. H

ow

man

y f

eet

did

th

e m

an

move i

n 3

seco

nd

s? H

ow

man

y f

eet

did

th

e s

had

ow

move i

n

3 s

eco

nd

s?

6

. T

he m

an

is

movin

g a

t a r

ate

of

5 f

eet/

seco

nd

. W

hat

rate

is

his

sh

ad

ow

movin

g?

How

does

this

rate

com

pare

to t

he c

on

ject

ure

you

mad

e i

n E

xerc

ise 1

? M

ak

e a

con

ject

ure

as

to

wh

y t

he r

esu

lts

are

lik

e t

his

.

20

ft

6 f

t

x f

t

5 f

t/sec

En

ric

hm

en

t7

-3

S

am

ple

an

sw

er:

His

sh

ad

ow

is

mo

vin

g m

ore

qu

ickly

th

an

he i

s.

4

0 f

t

5

7.1

4 f

t

5

5 f

t; 7

8.5

7 f

t

1

5 f

t; 2

1.4

3 f

t

7

.14 f

t /s;

Sam

ple

an

sw

er:

Th

e a

nsw

er

is c

on

sis

ten

t w

ith

th

e c

on

jectu

re

fro

m E

xerc

ise 1

. H

is s

had

ow

is g

ett

ing

lo

ng

er

as h

e m

oves f

urt

her

aw

ay

fro

m t

he l

am

pp

ost.



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

24

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Para

llel

Lin

es a

nd

Pro

po

rtio

nal

Part

s

Pro

po

rtio

na

l P

art

s w

ith

in T

ria

ng

les

In a

ny t

rian

gle

, a

lin

e p

ara

llel

to o

ne s

ide o

f a t

rian

gle

sep

ara

tes

the o

ther

two

sid

es

pro

port

ion

all

y.

Th

is i

s th

e T

rian

gle

Pro

port

ion

ali

ty T

heore

m.

Th

e c

on

vers

e i

s als

o t

rue.

If

"#

XY

‖

"#

RS

, th

en

RX

−

XT

= S

Y

−

YT

. If

RX

−

XT

= S

Y

−

YT

, th

en

"#

X

Y ‖

"#

R

S .

If X

an

d Y

are

th

e m

idp

oin

ts o

f −

−

RT

an

d −−

S

T ,

then

−−

XY

is

a

mid

se

gm

en

t of

the t

rian

gle

. T

he T

rian

gle

Mid

segm

en

t T

heore

m

state

s th

at

a m

idse

gm

en

t is

para

llel

to t

he t

hir

d s

ide a

nd

is

half

its

len

gth

.

If −

−

XY

is

a m

idse

gm

en

t, t

hen

"#

X

Y ‖

"#

R

S a

nd

XY

= 1

−

2 R

S.

YS

T

R

X

In

△A

BC

, −

−

EF

‖ −

−

CB

. F

ind

x.

F

C

BA

18

x+

22

x+

2

6E

Sin

ce −

−

EF

‖ −−

− C

B ,

AF

−

FB

= A

E

−

EC

.

x

+ 2

2

−

x +

2

= 1

8

−

6

6x +

132 =

18

x +

36

96 =

12x

8 =

x

In

△G

HJ

, H

K =

5,

KG

= 1

0, a

nd

JL

is o

ne

-ha

lf t

he

le

ng

th

of

−−

LG

. Is

−−

−

HK

‖ −

−

KL

?

Usi

ng t

he c

on

vers

e o

f th

e T

rian

gle

Pro

port

ion

ali

ty T

heore

m,

show

th

at

HK

−

KG

= J

L

−

LG

.

Let

JL

= x

an

d L

G =

2 x

.

HK

−

KG

= 5

−

10 =

1 −

2

JL

−

LG

= x

−

2x =

1 −

2

Sin

ce 1

−

2 =

1 −

2 ,

the s

ides

are

pro

port

ion

al

an

d

−−−

HJ

‖ −

−

KL

.

Exerc

ises

ALG

EB

RA

F

ind

th

e v

alu

e o

f x.

1.

5

5x

7

2.

9

20

x

18

3.

35x

4.

30

10

24

x

5.

11

x+

12

x33

6.

30

10

x+

10

x

7-4

Exam

ple

1Exam

ple

2

710

17.5

812

5

Lesson 7-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

25

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Para

llel

Lin

es a

nd

Pro

po

rtio

nal

Part

s

Pro

po

rtio

na

l P

art

s w

ith

Pa

rall

el

Lin

es

Wh

en

th

ree o

r m

ore

para

llel

lin

es

cut

two t

ran

svers

als

, th

ey s

ep

ara

te t

he t

ran

svers

als

in

to p

rop

ort

ion

al

part

s. I

f th

e r

ati

o o

f th

e p

art

s is

1,

then

th

e p

ara

llel

lin

es

sep

ara

te t

he t

ran

svers

als

in

to c

on

gru

en

t p

art

s.

If

ℓ1 ‖

ℓ2 ‖

ℓ3,

If

ℓ4 ‖

ℓ5 ‖

ℓ6 a

nd

th

en

a −

b =

c −

d .

u −

v =

1,

then

w

−

x

= 1

.

R

efe

r t

o l

ine

s ℓ

1,

ℓ2,

an

d ℓ

3 a

bo

ve

. If

a =

3,

b =

8,

an

d c

= 5

, fi

nd

d.

ℓ 1 ‖

ℓ2 ‖

ℓ3 s

o 3

−

8 =

5 −

d .

Th

en

3d

= 4

0 a

nd

d =

13 1

−

3 .

Exerc

ises

ALG

EB

RA

F

ind

x a

nd

y.

1

.

x+

12

12

3x

5x

2

. 2

x-

6

x+

3

12

3

.

2x

+ 4

3x

- 1

2y

+ 2

3y

4.

5

8y

+ 2

y

5

. 3

4

x+

4x

6

. 1

63

2

y+

3y

a

bd

uv

xw

ct

n

m

sℓ 1

ℓ 4ℓ 5

ℓ 6

ℓ 2 ℓ 3

7-4 Exam

ple

x =

8

x =

9

x =

5;

y =

2

y =

3 1

−

3

x =

12

y =

3



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

26

G

len

co

e G

eo

me

try

1. If

JK

= 7

, K

H =

21,

an

d J

L =

6,

2

. If

RU

= 8

, U

S =

14,

TV

= x

- 1

,

fin

d L

I.

an

d V

S =

17.5

, fi

nd

x a

nd

TV

.

De

term

ine

wh

eth

er −

−

BC

‖ −

−

DE

. J

usti

fy y

ou

r a

nsw

er.

3. A

D =

15,

DB

= 1

2,

AE

= 1

0,

an

d E

C =

8

yes;

AD

−

DB

= A

E

−

EC

= 5

−

4

4. B

D =

9,

BA

= 2

7,

an

d C

E =

1

−

3 E

A

n

o;

BD

−

DA

≠ C

E

−

EA

5. A

E =

30,

AC

= 4

5,

an

d A

D =

2D

B

yes;

AE

−

EC

= A

D

−

DB

= 2

−

1

−−

JH

is a

mid

se

gm

en

t o

f △

KL

M.

Fin

d t

he

va

lue

of

x.

6.

30

x

7.

9

x

8.

8

x

ALG

EB

RA

F

ind

x a

nd

y.

9

.

2x

+ 1

3y

- 8

y+

5x

+ 7

1

0.

3 – 2x

+ 2

x+

3

2y

- 1

3y

- 5

CB

ED

A

R

UV

T

S

HL

KJ

I

1

8

11;

10

x =

6,

y =

6.5

x =

2,

y =

4

Sk

ills

Practi

ce

Para

llel

Lin

es a

nd

Pro

po

rtio

nal

Part

s

7-4

15

18

8

Lesson 7-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

27

G

len

co

e G

eo

me

try

Practi

ce

Para

llel

Lin

es a

nd

Pro

po

rtio

nal

Part

s

1

. If

AD

= 2

4,

DB

= 2

7,

an

d E

B =

18,

2

. If

QT

= x

+ 6

, S

R =

12,

PS

= 2

7,

an

d

fin

d C

E.

T

R =

x -

4,

fin

d Q

T a

nd

TR

.

De

term

ine

wh

eth

er −

−

JK

‖ −

−−

NM

. J

usti

fy y

ou

r a

nsw

er.

3

. J

N =

18,

JL

= 3

0,

KM

= 2

1,

an

d M

L =

35

4

. K

M =

24,

KL

= 4

4,

an

d N

L =

5

−

6 J

N

−−

JH

is a

mid

se

gm

en

t o

f △

KL

M.

Fin

d t

he

va

lue

of

x.

5.

22

x

6

.

x+

7

x

7

. F

ind

x a

nd

y.

8. F

ind

x a

nd

y.

3x

- 4

2 – 3y

+ 3

1 – 3y

+ 6

5 – 4x

+ 3

4 – 3y

+ 2

3x

- 4

x+

1

4y

- 4

9

. M

AP

S O

n a

map

, W

ilm

ingto

n S

treet,

Beech

Dri

ve,

an

d A

sh G

rove

Lan

e a

pp

ear

to a

ll b

e p

ara

llel.

Th

e d

ista

nce

fro

m W

ilm

ingto

n t

o

Ash

Gro

ve a

lon

g K

en

dall

is

820 f

eet

an

d a

lon

g M

agn

oli

a,

660 f

eet.

If t

he d

ista

nce

betw

een

Beech

an

d A

sh G

rove a

lon

g M

agn

oli

a i

s

280 f

eet,

wh

at

is t

he d

ista

nce

betw

een

th

e t

wo s

treets

alo

ng

Ken

dall

?

Ash

Gro

ve

Bee

ch

Mag

no

liaK

end

all

Wilm

ing

ton

NL

J

KM

S

Q

T

RP

D

EC

BA

7-4

1

6

QT =

18;

TR

= 8

no

; 1

8

−

12 ≠

21

−

35

yes;

24

−

20 =

6

−

5

x

= 4

, y =

9x =

5

−

2 ,

y =

9

−

4

ab

ou

t 348 f

t

1

1

7



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

28

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Para

llel

Lin

es a

nd

Pro

po

rtio

nal

Part

s

Mee

eeow

ww

!

30 f

t

40 f

t

46 f

tx

8 f

t8 f

t

10 f

t

24 ft

10 f

t

x

Cay

St.

1 k

m1.4

km

0.8

km

Bay

St.

Dale St.

Earl S

t.

(not

dra

wn t

o s

cale

)

x

1

. C

AR

PE

NT

RY

Jak

e i

s

fixin

g a

n A

-fra

me.

He w

an

ts t

o a

dd

a

hori

zon

tal

sup

port

beam

half

way u

p

an

d p

ara

llel

to

the g

rou

nd

. H

ow

lon

g s

hou

ld t

his

beam

be?

2

. S

TR

EE

TS

In

th

e d

iagra

m,

Cay S

treet

an

d B

ay S

treet

are

para

llel.

Fin

d x

.

3

. JU

NG

LE

GY

MS

P

rass

ad

is

bu

ild

ing a

two-s

tory

ju

ngle

gym

acc

ord

ing t

o t

he

pla

ns

show

n.

Fin

d x

.

4

. FIR

EM

EN

A

cat

is s

tuck

in

a t

ree

an

d f

irem

en

try

to

resc

ue i

t. B

ase

d

on

th

e f

igu

re,

if a

fire

man

cli

mbs

to t

he t

op

of

the

lad

der,

how

far

aw

ay i

s th

e c

at?

5

. E

QU

AL P

AR

TS

N

ick

has

a s

tick

th

at

he

wou

ld l

ike t

o d

ivid

e i

nto

9 e

qu

al

part

s.

He p

lace

s it

on

a p

iece

of

gri

d p

ap

er

as

show

n.

Th

e g

rid

pap

er

is r

ule

d s

o t

hat

vert

ical

an

d h

ori

zon

tal

lin

es

are

equ

all

y

space

d.

a.

Exp

lain

how

he c

an

use

th

e g

rid

pap

er

to h

elp

him

fin

d w

here

he

need

s to

cu

t th

e s

tick

.

b.

Su

pp

ose

Nic

k w

an

ts t

o d

ivid

e h

is

stic

k i

nto

5 e

qu

al

part

s u

tili

zin

g t

he

gri

d p

ap

er.

Wh

at

can

he d

o?

8 f

eet

7-4 4

ft

1.1

2 k

m

13 f

t 4 i

n.

34.5

ft

B

ecau

se h

e p

osit

ion

ed

th

e

sti

ck s

o t

hat

its e

nd

s a

re

tou

ch

ing

ho

rizo

nta

l lin

es t

hat

are

9 u

nit

s a

part

, h

e c

an

cu

t

the s

tick w

here

ver

oth

er

ho

rizo

nta

l lin

es i

nte

rsect

it.

H

e c

an

ro

tate

th

e s

tick s

o i

t

tou

ch

es h

ori

zo

nta

l lin

es 5

un

its

ap

art

an

d a

pp

ly t

he s

am

e

meth

od

.

Lesson 7-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

29

G

len

co

e G

eo

me

try

En

ric

hm

en

t

Para

llel

Lin

es a

nd

Co

ng

ruen

t P

art

s

Th

ere

is

a t

heore

m s

tati

ng t

hat

if t

hre

e p

ara

llel

lin

es

cut

off

con

gru

en

t se

gm

en

ts o

n o

ne

tran

svers

al,

th

en

th

ey c

ut

off

con

gru

en

t se

gm

en

ts o

n a

ny t

ran

svers

al.

Th

is c

an

be s

how

n

for

an

y n

um

ber

of

para

llel

lin

es.

Th

e f

oll

ow

ing d

raft

ing t

ech

niq

ue u

ses

this

fact

to d

ivid

e a

segm

en

t in

to c

on

gru

en

t p

art

s.

−−

AB

to b

e s

ep

ara

ted

in

to f

ive c

on

gru

en

t p

art

s. T

his

can

be

don

e v

ery

acc

ura

tely

wit

hou

t u

sin

g a

ru

ler.

All

th

at

is n

eed

ed

is a

com

pass

an

d a

pie

ce o

f n

ote

book

pap

er.

Ste

p 1

H

old

th

e c

orn

er

of

a p

iece

of

note

book

pap

er

at

poin

t A

.

Ste

p 2

F

rom

poin

t A

, d

raw

a s

egm

en

t alo

ng t

he p

ap

er

that

is f

ive s

pace

s lo

ng.

Mark

wh

ere

th

e l

ines

of

the n

ote

book

pap

er

meet

the s

egm

en

t. L

abel

the f

ifth

poin

t P

.

Ste

p 3

D

raw

−−

PB

. T

hro

ugh

each

of

the o

ther

mark

s

on

−−

AP

, co

nst

ruct

a l

ine p

ara

llel

to −

−

BP

. T

he

poin

ts w

here

th

ese

lin

es

inte

rsect

−−

AB

wil

l

div

ide −

−

AB

in

to f

ive c

on

gru

en

t se

gm

en

ts.

Use

a c

om

pa

ss a

nd

a p

iece

of

no

teb

oo

k p

ap

er t

o d

ivid

e

ea

ch

se

gm

en

t in

to t

he

giv

en

nu

mb

er o

f co

ng

ru

en

t p

arts

.

1

. si

x c

on

gru

en

t p

art

s

2

. se

ven

con

gru

en

t p

art

s

AB

AB

A

P

B

AB

P

AB

AB

7-4

See s

tud

en

ts’

wo

rk.



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

30

G

len

co

e G

eo

me

try

7-5

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Part

s o

f S

imilar

Tri

an

gle

s

Sp

eci

al

Se

gm

en

ts o

f S

imil

ar

Tri

an

gle

s W

hen

tw

o t

rian

gle

s are

sim

ilar,

co

rresp

on

din

g a

ltit

ud

es,

an

gle

bis

ect

ors

, an

d m

ed

ian

s are

pro

port

ion

al

to t

he c

orr

esp

on

din

g

sid

es.

Exerc

ises

Fin

d x

.

1

. x

36

18

20

2

.

69

12

x

3

.

3

3

4

x

4

.

10

10

x

8

87

5

.

45

42

x

30

6

.

14

12

12

x

Exam

ple

In

th

e f

igu

re, △ABC

∼ △

XYZ

, w

ith

an

gle

bis

ecto

rs a

s s

ho

wn

. F

ind

x.

Sin

ce △

AB

C ∼

△X

YZ

, th

e m

easu

res

of

the a

ngle

bis

ect

ors

are

pro

port

ion

al

to

the m

easu

res

of

a p

air

of

corr

esp

on

din

g s

ides.

AB

−

XY

= B

D

−

WY

24

−

x

= 1

0

−

8

10x =

24(8

)

10

x =

19

2

x =

19.2

24

10 D

W

Y

ZX

C

B

A

8

x

10

8

2.2

58.7

5

14

28


NA

ME

DA

TE

PE

RIO

D

Lesson 7-5

Ch

ap

ter

7

31

G

len

co

e G

eo

me

try

7-5

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Part

s o

f S

imilar

Tri

an

gle

s

Tri

an

gle

An

gle

Bis

ect

or

Th

eo

rem

A

n a

ngle

bis

ect

or

in a

tri

an

gle

sep

ara

tes

the

op

posi

te s

ide i

nto

tw

o s

egm

en

ts t

hat

are

pro

port

ion

al

to t

he l

en

gth

s of

the o

ther

two s

ides.

F

ind

x.

Sin

ce −

−−

SU

is

an

an

gle

bis

ect

or,

RU

−

TU

= R

S

−

TS

.

x

−

20 =

15

−

30

30x =

20(1

5)

30x =

30

0

x =

10

Exerc

ises

Fin

d t

he

va

lue

of

ea

ch

va

ria

ble

.

1

. 25

20

28

x

2.

15

9

3a

3

.

36

42

46.8

n

4.

10

15

13

x

5

.

110

100

160

r

6

.

17

11

x+

7

x

Exam

ple

x2

0

30

15

UT

S

R

35

5

66

12 5

−

6

25.2

26



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

32

G

len

co

e G

eo

me

try

7-5

Sk

ills

Practi

ce

Part

s o

f S

imilar

Tri

an

gle

sF

ind

x.

1.

2.

7 7

22

x5.2

5

5.2

5

153

3

10x

3.

4.

20

10

x

22

18

121

2.6

x

5. If

△R

ST

∼ △

EF

G,

−−

−

SH

is

an

6. If

△A

BC

∼ △

MN

P,

−−

−

AD

is

an

alt

itu

de o

f △

RS

T,

−−

FJ

is

an

alt

itu

de o

f

alt

itu

de o

f △

AB

C,

−−

−

MQ

is

an

alt

itu

de o

f

△E

FG

, S

T =

6,

SH

= 5

, an

d F

J =

7,

△M

NP

, A

B =

24,

AD

= 1

4,

an

dfi

nd

FG

. M

Q =

10.5

, fi

nd

MN

.

JG

E

F

HT

R

S

C

A

BD

PN

QM

Fin

d t

he

va

lue

of e

ach

va

ria

ble

.

7.

8.

26

12

8

m

7

20

24

x

8.4

18

16.5

22

17 1

−

3

8.4

11

17

.1


NA

ME

DA

TE

PE

RIO

D

Lesson 7-5

Ch

ap

ter

7

33

G

len

co

e G

eo

me

try

Practi

ce

Part

s o

f S

imilar

Tri

an

gle

s

ALG

EB

RA

F

ind

x.

1

.

32

30

24

x

2.

26

25

39

x

3

.

40

2x

+ 1

25

x+

4

4.

5

. If

△J

KL

∼△

NP

R,

−−

−K

M i

s an

6

. If

△S

TU

∼△

XY

Z,

−−

−U

A i

s an

alt

itu

de o

f △

JK

L,

−−

PT

is

an

alt

itu

de

alt

itu

de o

f △

ST

U,

−−

ZB

is

an

alt

itu

de

of

△N

PR

, K

L=

28,

KM

= 1

8,

an

d

of

△X

YZ

, U

T=

8.5

, U

A=

6,

an

d

PT

= 1

5.7

5,

fin

d P

R.

Z

B=

11.4

, fi

nd

ZY

.

MJ

L

K

TN

R

P

YX

BZ

TS

AU

7

. P

HO

TO

GR

AP

HY

F

ran

cin

e h

as

a c

am

era

in

wh

ich

th

e d

ista

nce

fro

m t

he l

en

s to

th

e f

ilm

is

24 m

illi

mete

rs.

a.

If F

ran

cin

e t

ak

es

a f

ull

-len

gth

ph

oto

gra

ph

of

her

frie

nd

fro

m a

dis

tan

ce o

f 3 m

ete

rs

an

d t

he h

eig

ht

of

her

frie

nd

is

140 c

en

tim

ete

rs,

wh

at

wil

l be t

he h

eig

ht

of

the i

mage

on

th

e f

ilm

? (H

int:

Con

vert

to t

he s

am

e u

nit

of

measu

re.)

b.

Su

pp

ose

th

e h

eig

ht

of

the i

mage o

n t

he f

ilm

of

her

frie

nd

is

15 m

illi

mete

rs.

If

Fra

nci

ne t

ook

a f

ull

-len

gth

sh

ot,

wh

at

was

the d

ista

nce

betw

een

th

e c

am

era

an

d h

er

frie

nd

?

7-5

x

28

20

30

22.5

16.7

24.5

16.1

5

13.5

16.8

11.2

mm

2.2

4 m



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

34

G

len

co

e G

eo

me

try

Wo

rd

Pro

ble

m P

racti

ce

Part

s o

f S

imilar

Tri

an

gle

s

1

. FLA

GS

An

oce

an

lin

er

is f

lyin

g t

wo

sim

ilar

tria

ngu

lar

flags

on

a f

lag p

ole

.

Th

e a

ltit

ud

e o

f th

e l

arg

er

flag i

s th

ree

tim

es

the a

ltit

ud

e o

f th

e s

mall

er

flag.

If

the m

easu

re o

f a l

eg o

n t

he l

arg

er

flag i

s

45 i

nch

es,

fin

d t

he m

easu

re o

f th

e

corr

esp

on

din

g l

eg o

n t

he s

mall

er

flag.

2

. T

EN

TS

Jan

a w

en

t ca

mp

ing a

nd

sta

yed

in a

ten

t sh

ap

ed

lik

e a

tri

an

gle

. In

a

ph

oto

of

the t

en

t, t

he b

ase

of

the t

en

t is

6 i

nch

es

an

d t

he a

ltit

ud

e i

s 5 i

nch

es.

Th

e a

ctu

al

base

was

12 f

eet

lon

g.

Wh

at

was

the h

eig

ht

of

the a

ctu

al

ten

t?

x

3

. P

LA

YG

RO

UN

D T

he p

laygro

un

d a

t

Han

k’s

sch

ool

has

a l

arg

e r

igh

t tr

ian

gle

pain

ted

in

th

e g

rou

nd

. H

an

k s

tart

s

at

the r

igh

t an

gle

corn

er

an

d w

alk

s

tow

ard

th

e o

pp

osi

te s

ide a

lon

g a

n a

ngle

bis

ect

or

an

d s

top

s w

hen

he g

ets

to t

he

hyp

ote

nu

se.

A

B

45˚

30

ft

40 f

t

50 ft

How

mu

ch f

art

her

from

Han

k i

s p

oin

t B

vers

us

poin

t A

?

4. FLA

G P

OLE

S A

fla

g p

ole

att

ach

ed

to t

he

sid

e o

f a b

uil

din

g i

s su

pp

ort

ed

wit

h

a n

etw

ork

of

stri

ngs

as

show

n i

n t

he

figu

re.

AB

C

DF E

Th

e r

iggin

g i

s d

on

e s

o t

hat

AE

=E

F,

AC

=C

D,

an

d A

B=

BC

. W

hat

is t

he

rati

o o

f C

F t

o B

E?

5

. C

OP

IES

G

ord

on

mad

e a

ph

oto

cop

y o

f a

page f

rom

his

geom

etr

y b

ook

to e

nla

rge

on

e o

f th

e f

igu

res.

Th

e a

ctu

al

figu

re t

hat

he c

op

ied

is

show

n b

elo

w.

39 m

m

Median

Altitude

30 m

m

29

mm

Th

e p

hoto

cop

y c

am

e o

ut

poorl

y.

Gord

on

cou

ld n

ot

read

th

e n

um

bers

on

th

e

ph

oto

cop

y,

alt

hou

gh

th

e t

rian

gle

its

elf

was

clear.

Gord

on

measu

red

th

e b

ase

of

the e

nla

rged

tri

an

gle

an

d f

ou

nd

it

to b

e

200 m

illi

mete

rs.

a

. W

hat

is t

he l

en

gth

of

the d

raw

n

alt

itu

de o

f th

e e

nla

rged

tri

an

gle

?

Rou

nd

you

r an

swer

to t

he n

eare

st

mil

lim

ete

r.

b

. W

hat

is t

he l

en

gth

of

the d

raw

n

med

ian

of

the e

nla

rged

tri

an

gle

?

Rou

nd

you

r an

swer

to t

he n

eare

st

mil

lim

ete

r.

7-5 1

5 i

nch

es

10 f

t

7 1

−

7 f

t

2 :1 1

49 m

m

154 m

m

Lesson 7-5


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

35

G

len

co

e G

eo

me

try

A P

roo

f o

f P

yth

ag

ore

an

Th

eo

rem

1. F

or

righ

t tr

ian

gle

AB

C w

ith

rig

ht

an

gle

C,

an

d

alt

itu

de −

−−

CD

as

show

n a

t th

e r

igh

t, n

am

e t

hre

e

sim

ilar

tria

ngle

s.

2

. L

ist

the t

hre

e s

imil

ar

tria

ngle

s as

head

ings

in

the t

able

belo

w.

Use

th

e f

igu

re t

o c

om

ple

te t

he

table

to l

ist

the c

orr

esp

on

din

g p

art

s of

the t

hre

e

sim

ilar

righ

t tr

ian

gle

s.

△A

BC

△A

CD

△C

BD

Sh

ort

Le

ga

hd

Lo

ng

Le

gb

eh

Hy

po

ten

us

ec

ba

3

. U

se t

he c

orr

esp

on

din

g p

art

s of

these

sim

ilar

tria

ngle

s an

d t

heir

pro

port

ion

s

to c

om

ple

te t

he s

tate

men

ts i

n t

he p

roof

an

d a

lgebra

icall

y p

rove t

he

Pyth

agore

an

Th

eore

m.

S

ta

te

me

nts

Re

aso

ns

1.

Rig

ht

tria

ngle

AB

C w

ith

1

. G

iven

alt

itu

de −

−−

CD

.

2

.

2.

Corr

esp

on

din

g p

art

s of

sim

ilar

tria

ngle

s are

in

th

e s

am

e r

ati

o.

3

.

3.

Cro

ss P

rod

uct

s P

rop

ert

y

4

.

4.

Ad

dit

ion

Pro

pert

y o

f E

qu

ali

ty

5

.

5.

Su

bst

itu

tion

6

.

6.

Dis

trib

uti

ve P

rop

ert

y

7

.

7.

Segm

en

t ad

dit

ion

8

. a

2 +

b2 =

c2

8.

Su

bst

itu

tion

A

ab

h

de

c

B

C D

En

ric

hm

en

t7

-5

△

AB

C ∼

△A

CD

∼ △

CB

D

a2 =

cd

, b

2 =

ce

a2 +

b2 =

cd

+ b

2

a2 +

b2 =

cd

+ c

e

a2 +

b2 =

c(d

+ e

)

d +

e =

c

a

−

c =

d

−

a ,

b

−

c =

e

−

b



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

36

G

len

co

e G

eo

me

try

You

can

use

a s

pre

ad

sheet

to c

reate

a S

ierp

insk

i tr

ian

gle

.

U

se

a s

pre

ad

sh

ee

t to

cre

ate

a S

ierp

insk

i tr

ian

gle

.

Ste

p 1

In

cell

A2,

en

ter

1.

In c

ell

A3,

en

ter

an

equ

als

sig

n f

oll

ow

ed

by A

2 +

1 . T

his

wil

l re

turn

th

e n

um

ber

of

itera

tion

s. C

lick

on

th

e b

ott

om

rig

ht

corn

er

of

cell

A2 a

nd

d

rag i

t th

rou

gh

cell

A500 t

o g

et

500 i

tera

tion

s.

Ste

p 2

In

cell

B1,

en

ter

0.5

. T

his

in

dic

ate

s th

e m

idp

oin

ts o

f th

e s

egm

en

ts.

Ste

p 3

In

cell

C1,

en

ter

1 a

nd

in

cell

D1,

en

ter

0.3

.

Ste

p 4

In

cell

B2,

en

ter

an

equ

als

sig

n f

oll

ow

ed

by 3

*R

AN

D()

. C

lick

on

th

e b

ott

om

rig

ht

corn

er

of

cell

B2 a

nd

dra

g i

n t

hro

ugh

cell

B500 t

o g

et

500 i

tera

tion

s.

Ste

p 5

In

cell

C2,

en

ter

an

equ

als

sig

n f

oll

ow

ed

by t

he r

ecu

rsiv

e f

orm

ula

IF

(B2

<1,(

12

$B

$1)*

C1,I

F(B

2<

2,(

1-

$B

$1)*

C1

+$B

$1,(

1-

$B

$1)*

C1

+2

*$B

$1))

. T

his

w

ill

retu

rn t

he x

valu

es

of

the p

oin

ts t

o b

e g

rap

hed

. C

lick

on

th

e b

ott

om

rig

ht

corn

er

of

cell

C2 a

nd

dra

g t

hro

ugh

cell

C500.

Ste

p 6

In

cell

D2,

en

ter

an

equ

als

sig

n f

oll

ow

ed

by t

he r

ecu

rsiv

e f

orm

ula

IF

(B2

<1,(

1-

$B

$1)*

D1,I

F(B

2<

2,(

1-

$B

$1)*

D1

+$B

$1,(

1-

$B

$1)*

D1))

. T

his

wil

l re

turn

th

e y

valu

es

of

the p

oin

ts t

o b

e g

rap

hed

. C

lick

on

th

e b

ott

om

rig

ht

corn

er

of

cell

D2 a

nd

dra

g t

hro

ugh

ce

ll D

500.

Ste

p 7

T

o g

rap

h t

he v

alu

es

in

colu

mn

s C

an

d D

, fi

rst

hig

hli

gh

t all

of

the d

ata

in

th

e t

wo c

olu

mn

s. N

ext,

ch

oose

th

e c

hart

wiz

ard

fr

om

th

e t

oolb

ar.

Sele

ct X

Y

(Sca

tter)

. P

ress

Next,

Next.

T

hen

sele

ct t

he G

rid

lin

es

tab a

nd

un

check

th

e M

ajo

r gri

dli

nes.

Th

en

pre

ss N

ext

an

d F

inis

h.

Th

is w

ill

retu

rn t

he S

ierp

insk

i tr

ian

gle

.

Exerc

ises

An

aly

ze

yo

ur d

ra

win

g.

1

. W

hat

hap

pen

s to

you

r d

raw

ing i

f you

have m

ore

ite

rati

on

s? T

ry 1

000,

2000,

an

d 5

000.

2

. C

han

ge t

he 0

.5 t

o 2

/3 i

n c

ell

B1.

(Hin

t: Y

ou

may n

eed

to e

nte

r 0.6

66666 f

or

2/3

.) H

ow

d

oes

this

ch

an

ge t

he p

ictu

re?

3

. C

han

ge 0

.3 t

o 0

.6 i

n c

ell

D1.

How

does

this

ch

an

ge t

he d

raw

ing?

4

. E

nte

r th

e f

oll

ow

ing i

nto

a n

ew

sp

read

sheet

an

d d

esc

ribe w

hat

you

see.

S

tep

1:

In c

ell

A2, en

ter

the f

orm

ula

= A

1.

S

tep

2:

In c

ell

B2, en

ter

the f

orm

ula

= A

1 +

B1.

S

tep

3:

Cli

ck o

n t

he b

ott

om

rig

ht

corn

er

of

cell

B2 a

nd

dra

g t

hro

ugh

cell

L2.

S

tep

4:

Fir

st, cl

ick

on

th

e 2

next

to c

ell

A2. T

hen

, cl

ick

on

th

e b

ott

om

left

corn

er

of

2

an

d d

rag d

ow

n t

hro

ugh

cell

12.

S

tep

5:

In c

ell

A1, en

ter

a 1

an

d p

ress

EN

TE

R.

Sh

ee

t 1

Sh

ee

t 2

Sh

ee

t 3

Th

e p

ictu

re l

oo

ks s

harp

er

wit

h m

ore

ite

rati

on

s.

See s

tud

en

ts’

wo

rk.

See s

tud

en

ts’

wo

rk.

Pascal’s t

rian

gle

Sp

read

sheet

Act

ivit

y

Fra

cta

ls

7-5 Exam

ple

Lesson 7-6


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

37

G

len

co

e G

eo

me

try

7-6

Ide

nti

fy S

imil

ari

ty T

ran

sfo

rma

tio

ns

A d

ila

tio

n i

s a t

ran

sform

ati

on

th

at

en

larg

es

or

red

uce

s th

e o

rigin

al

figu

re p

rop

ort

ion

all

y.

Th

e s

ca

le f

acto

r o

f a

dil

ati

on

, k,

is t

he r

ati

o

of

a l

en

gth

on

th

e i

mage t

o a

corr

esp

on

din

g l

en

gth

on

th

e p

reim

age.

A d

ilati

on

wit

h k

> 1

is

an

en

larg

em

en

t. A

dil

ati

on

wit

h 0

< k

< 1

is

a r

ed

ucti

on

.

D

ete

rm

ine

wh

eth

er t

he

dil

ati

on

fro

m A

to

B i

s a

n enlargement

or a

reduction

. T

he

n f

ind

th

e s

ca

le f

acto

r o

f th

e d

ila

tio

n.

Exerc

ises

De

term

ine

wh

eth

er t

he

dil

ati

on

fro

m A

to

B i

s a

n enlargement

or a

reduction

.

Th

en

fin

d t

he

sca

le f

acto

r o

f th

e d

ila

tio

n.

1.

y

x

2.

y

x

3

. y

x

4.

y

x

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Sim

ilari

ty T

ran

sfo

rmati

on

s

Exam

ple

a.

y

x

B i

s la

rger

than

A,

so t

he d

ilati

on

is

an

en

larg

em

en

t.

Th

e d

ista

nce

betw

een

th

e v

ert

ices

at

(-3,

4)

an

d (

-1,

4)

for

A i

s 2.

Th

e

dis

tan

ce b

etw

een

th

e v

ert

ices

at

(0,

3)

an

d (

4,

3)

for

B i

s 4.

Th

e s

cale

fact

or

is 4

−

2 o

r 2.

b.

y

x

B i

s sm

all

er

than

A,

so t

he d

ilati

on

is

a

red

uct

ion

.

Th

e d

ista

nce

betw

een

th

e v

ert

ices

at

(2,

3)

an

d

(2,

-3)

for

A i

s 6.

Th

e d

ista

nce

betw

een

th

e

vert

ices

at

(2,

1)

an

d (

2,

-2)

for

B i

s 3.

Th

e s

cale

fact

or

is 3

−

6 o

r 1

−

2 .

e

nla

rgem

en

t; 2

r

ed

ucti

on

; 1

−

2

r

ed

ucti

on

; 4

−

5

red

ucti

on

; 1

−

2

Answers (Lesson 7-5 and Lesson 7-6)


Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

38

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Sim

ilari

ty T

ran

sfo

rmati

on

s

Ve

rify

Sim

ila

rity

Y

ou

can

veri

fy t

hat

a d

ilati

on

pro

du

ces

a s

imil

ar

figu

re b

y c

om

pari

ng

the r

ati

os

of

all

corr

esp

on

din

g s

ides.

For

tria

ngle

s, y

ou

can

als

o u

se S

AS

Sim

ilari

ty.

G

ra

ph

th

e o

rig

ina

l fi

gu

re

an

d i

ts d

ila

ted

im

ag

e.

Th

en

ve

rif

y t

ha

t th

e d

ila

tio

n i

s a

sim

ila

rit

y t

ra

nsfo

rm

ati

on

.

Exam

ple

a.

orig

ina

l: A

(-3

, 4

), B

(2,

4),

C(-

3, -

4)

im

ag

e: D

(1,

0),

E(3

.5,

0),

F(1

, -

4)

Gra

ph

each

fig

ure

. S

ince

∠A

an

d ∠D

are

both

rig

ht

an

gle

s, ∠A

" ∠D

. S

how

th

at

the l

en

gth

s of

the s

ides

that

incl

ud

e

∠A

an

d ∠D

are

pro

port

ion

al

to p

rove

sim

ilari

ty b

y S

AS

.

Use

th

e c

oord

inate

gri

d t

o f

ind

th

e

len

gth

s of

vert

ical

segm

en

ts A

C a

nd

DF

an

d h

ori

zon

tal

segm

en

ts A

B a

nd

DE

.

AC

−

DF

= 8

−

4 =

2 a

nd

AB

−

DE

=

5

−

2.5

= 2

,

so A

C

−

DF

= A

B

−

DE

.

Sin

ce t

he l

en

gth

s of

the s

ides

that

incl

ud

e

∠A

an

d ∠D

are

pro

port

ion

al,

△A

BC

∼

△D

EF

by S

AS

sim

ilari

ty.

b.

orig

ina

l: G

(-4

, 1

), H

(0,

4),

J(4

, 1

), K

(0, -

2)

im

ag

e: L

(-2

, 1

.5),

M(0

, 3

), N

(2,

1.5

), P

(0,

0)

Use

th

e d

ista

nce

form

ula

to f

ind

th

e l

en

gth

of

each

sid

e.

GH

= √

''

'

42 +

32 =

√ '

'

25 =

5

HJ

= √

''

'

42 +

32 =

√ '

'

25 =

5

JK

= √

''

'

42 +

32 =

√ '

'

25 =

5

KG

= √

''

'

42 +

32 =

√ '

'

25 =

5

LM

= √

''

''

22 +

1.5

2 =

√ '

'

6.2

5 =

2.5

MN

= √

''

''

22 +

1.5

2 =

√ '

'

6.2

5 =

2.5

NP

= √

''

''

22 +

1.5

2 =

√ '

'

6.2

5 =

2.5

PL

= √

''

''

22 +

1.5

2 =

√ '

'

6.2

5 =

2.5

Fin

d a

nd

com

pare

th

e r

ati

os

of

corr

esp

on

din

g

sid

es.

GH

−

LM

=

5

−

2.5

= 2

, H

J

−

MN

=

5

−

2.5

= 2

,

JK

−

NP

=

5

−

2.5

= 2

, K

C

−

PL

=

5

−

2.5

= 2

.

Sin

ce G

H

−

LM

=

HJ

−

MN

= J

K

−

NP

= K

C

−

PL

, G

HJ

K ∼

LM

NP

.

y

x

y

x

Exerc

ises

Gra

ph

th

e o

rig

ina

l fi

gu

re

an

d i

ts d

ila

ted

im

ag

e.

Th

en

ve

rif

y t

ha

t th

e d

ila

tio

n i

s a

sim

ila

rit

y t

ra

nsfo

rm

ati

on

.

1

. A

(-4,

-3),

B(2

, 5),

C(2

, -

3),

2

. P

(-4,

1),

Q(-

2,

4),

R(0

, 1),

S(-

2,

-2),

D

(-2,

-2),

E(1

, 3),

F(1

, -

2)

W

(1,

-1.5

), X

(2,

0),

Y(3

, -

1.5

), Z

(2,

-3)

7-6

S

ee s

tud

en

ts’

wo

rk.


NA

ME

DA

TE

PE

RIO

D

Lesson 7-6

Ch

ap

ter

7

39

G

len

co

e G

eo

me

try

7-6

Dete

rm

ine w

heth

er t

he d

ila

tio

n f

ro

m A

to

B i

s a

n enlargement

or

a reduction

. T

hen

fin

d t

he s

ca

le f

acto

r o

f th

e d

ila

tio

n.

1.

y

x

2

. y

x

3.

y

x

4

. y

x

Gra

ph

th

e o

rig

ina

l fi

gu

re

an

d i

ts d

ila

ted

im

ag

e.

Th

en

ve

rif

y t

ha

t th

e d

ila

tio

n i

s a

sim

ila

rit

y t

ra

nsfo

rm

ati

on

.

5.

A(–

3,

4),

B(3

, 4),

C(–

3,

–2);

6

. F

(–3,

4),

G(2

, 4),

H(2

, –2),

J(–

3,

–2);

A'(–

2,

3),

B'(0

, 3

), C'(–

2,

1)

F'(–

1.5

, 3

), G'(1

, 3

), H'(1

, 0

), J'(–

1.5

, 0

)

Sk

ills

Pra

ctic

e

Sim

ilari

ty T

ran

sfo

rmati

on

s

′

y

x

′′

′ ′

y

x′′

7.

P(–

3,

1),

Q(–

1,

1),

R(–

1,

–3);

8

. A

(–5,

–1),

B(0

, 1),

C(5

, –1),

D(0

, –3);

P'(–

1,

4),

Q'(3

, 4

), R'(3

, –

4)

A'(1

, –

1.5

), B'(2

, 0

), C'(3

, –

1.5

), D'(2

, –

3)

′

′ ′

y

x

′ ′

′

y

x

′

e

nla

rgem

en

t; 4

−

3

en

larg

em

en

t; 7

−

3

r

ed

ucti

on

; 1

−

2

red

ucti

on

; 3

−

4

s

imilar

by S

AS

sim

ilar

by p

rop

ort

ion

al

sid

es

s

imilar

by S

AS

sim

ilar

by p

rop

ort

ion

al

sid

es



Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

40

G

len

co

e G

eo

me

try

7-6

Practi

ce

Sim

ilari

ty T

ran

sfo

rmati

on

sD

ete

rm

ine

wh

eth

er t

he

dil

ati

on

fro

m A

to

B i

s a

n enlargement

or a

reduction

.

Th

en

fin

d t

he

sca

le f

acto

r o

f th

e d

ila

tio

n.

1.

y

x

2

. y

x

e

nla

rgem

en

t; 2

red

ucti

on

; 1

−

2

3

. y

x

4

. y

x

e

nla

rgem

en

t; 3

red

ucti

on

; 3

−

4

5

. y

x

6

. y

x

e

nla

rgem

en

t; 7

−

5

re

du

cti

on

; 1

−

2

Gra

ph

th

e o

rig

ina

l fi

gu

re

an

d i

ts d

ila

ted

im

ag

e.

Th

en

ve

rif

y t

ha

t th

e d

ila

tio

n i

s a

sim

ila

rit

y t

ra

nsfo

rm

ati

on

.

7

. Q

(1,

4),

R(4

, 4),

S(4

, -

1),

8

. A

(-4,

2),

B(0

, 4),

C(4

, 2),

D(0

, -

2),

X

(-4,

5),

Y(2

, 5),

Z(2

, -

5)

F

(-2,

1),

G(0

, 2),

H(2

, 1),

J(0

, -

1)

9

. FA

BR

IC R

yan

bu

ys

an

8-f

oot-

lon

g b

y 6

-foot-

wid

e p

iece

of

fabri

c as

show

n.

He w

an

ts t

o

cut

a s

mall

er,

sim

ilar

rect

an

gu

lar

pie

ce t

hat

has

a s

cale

fact

or

of k =

1

−

4 .

If p

oin

t A

(-4,

3)

is t

he t

op

left

-han

d v

ert

ex o

f both

th

e o

rigin

al

pie

ce o

f fa

bri

c an

d t

he p

iece

Ryan

wis

hes

to c

ut

ou

t, w

hat

are

th

e c

oord

inate

s of

the v

ert

ices

for

the p

iece

Ryan

wil

l cu

t?

(

-2,

3),

(-

2,

1.5

), (

-4,

1.5

)

See s

tud

en

ts’

wo

rk.

S

imilar

by S

AS

Sim

ilar

by p

rop

ort

ion

al

sid

es


NA

ME

DA

TE

PE

RIO

D

Lesson 7-6

Ch

ap

ter

7

41

G

len

co

e G

eo

me

try

7-6

1

. C

ITY

PLA

NN

ING

T

he s

tan

dard

siz

e o

f a

city

blo

ck i

n M

an

hatt

an

is

264 f

eet

by

900 f

eet.

Th

e c

ity p

lan

ner

of

Mech

lin

bu

rg

wan

ts t

o b

uil

d a

new

su

bd

ivis

ion

usi

ng

sim

ilar

blo

cks

so t

he d

imen

sion

s of

a

stan

dard

Man

hatt

an

blo

ck a

re e

nla

rged

by 2

.5 t

imes.

Wh

at

wil

l be t

he n

ew

d

imen

sion

s of

each

en

larg

ed

blo

ck?

660 f

eet

by 2

250 f

eet

2

. S

TO

RA

GE

SH

ED

A

loca

l h

om

e

imp

rovem

en

t st

ore

sell

s d

iffe

ren

t si

zes

of

stora

ge s

hed

s. T

he m

ost

exp

en

sive s

hed

h

as

a f

ootp

rin

t th

at

is 1

5 f

eet

wid

e b

y 2

1

feet

lon

g.

Th

e l

east

exp

en

sive s

hed

has

a

footp

rin

t th

at

is 1

0 f

eet

wid

e b

y 1

4 f

eet

lon

g.

Are

th

e f

ootp

rin

ts o

f th

e t

wo s

hed

s si

mil

ar?

If

so,

tell

wh

eth

er

the f

ootp

rin

ts

of

the l

east

exp

en

sive s

hed

is

an

en

larg

em

en

t or

a r

ed

uct

ion

, an

d f

ind

th

e

scale

fact

or

from

th

e m

ost

exp

en

sive s

hed

to

th

e l

east

exp

en

sive s

hed

.

yes;

red

ucti

on

; 2

−

3

3

. FIN

D T

HE

ER

RO

R Jere

my a

nd

Eli

sa a

re

con

stru

ctin

g d

ilati

on

s of

rect

an

gle

ABCD

for

their

geom

etr

y c

lass

. Jere

my d

raw

s an

en

larg

em

en

t FGHJ

th

at

con

tain

s th

e

poin

ts F

(-

3,

3)

an

d J

(3,

-6).

Eli

sa

dra

ws

a r

ed

uct

ion

KLMN

th

at

con

tain

s th

e p

oin

ts L

(-3,

3)

an

d N

(-2,

2).

Wh

ich

p

ers

on

mad

e a

n e

rror

in t

heir

dil

ati

on

? E

xp

lain

.

y

x

Elisa m

ad

e t

he e

rro

r; K

LM

N i

s n

ot

sim

ilar

to t

he o

rig

inal

AB

CD

.

4

. B

AN

NE

RS

T

he B

aysi

de H

igh

Sch

ool

Sp

irit

Squ

ad

is

mak

ing a

ban

ner

to t

ak

e

to a

way g

am

es.

Th

e b

an

ner

they u

se f

or

hom

e g

am

es

is s

how

n b

elo

w.

If t

he n

ew

ban

ner

is t

o b

e a

red

uct

ion

of

the h

om

e

gam

e b

an

ner

wit

h a

sca

le f

act

or

of

1 − 2 ,

wh

at

wil

l be t

he h

eig

ht

of

the n

ew

ban

ner?

13 ft

3.5

ft

1.7

5 f

t

5

. R

EA

SO

NIN

G C

on

sid

er

the i

mage QRST

of

a r

ect

an

gle

WXYZ

.

a.

Is i

t p

oss

ible

th

at

poin

t Q

an

d p

oin

t W

cou

ld h

ave t

he s

am

e c

oord

inate

s? I

f so

, w

hat

mu

st b

e t

rue a

bou

t p

oin

t Q

?

yes,

po

int

Q i

s t

he c

en

ter

of

dilati

on

.

b.

Is i

t p

oss

ible

th

at

both

poin

ts R

an

d X

cou

ld h

ave t

he s

am

e c

oord

inate

s if

p

oin

ts T

an

d Z

have t

he s

am

e

coord

inate

s? I

f so

, w

hat

are

th

e

poss

ible

valu

es

of

the s

cale

fact

or k f

or

the d

ilati

on

?

yes;

k =

1

Wo

rd

Pro

ble

m P

racti

ce

Sim

ilari

ty T

ran

sfo

rmati

on

s



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

42

G

len

co

e G

eo

me

try

7-6

En

rich

men

t

Med

ial

an

d O

rth

ic T

rian

gle

sT

he m

ed

ial

tria

ng

le i

s th

e t

rian

gle

form

ed

by c

on

nect

ing t

he

mid

poin

ts o

f each

sid

e o

f th

e t

rian

gle

. T

he t

rian

gle

form

ed

by

A’, B

’, a

nd

C’ is

th

e m

ed

ial

tria

ngle

of

tria

ngle

AB

C.

Use

a r

ule

r a

nd

co

mp

ass.

Dra

w t

he

me

dia

l tr

ian

gle

fo

r e

ach

tria

ng

le b

elo

w.

1

.

2.

3. U

se t

he t

rian

gle

s you

have c

reate

d t

o s

how

th

at

the m

ed

ial

tria

ngle

is

sim

ilar

to t

he

ori

gin

al

tria

ngle

.

S

ee s

tud

en

ts’

wo

rk.

Th

e o

rth

ic t

ria

ng

le i

s th

e t

rian

gle

form

ed

by c

on

nect

ing t

he

en

dp

oin

ts o

f each

of

the a

ltit

ud

es

of

the t

rian

gle

. T

he t

rian

gle

form

ed

by F

, D

, an

d,

E i

s th

e o

rth

ic t

rian

gle

of

tria

ngle

AB

C.

Use

a r

ule

r a

nd

co

mp

ass.

Dra

w t

he

orth

ic t

ria

ng

le f

or e

ach

tria

ng

le b

elo

w.

1

.

2.

3

. U

se t

he t

rian

gle

s you

have c

reate

d t

o s

how

th

at

the o

rth

ic t

rian

gle

is

sim

ilar

to t

he

ori

gin

al

tria

ngle

.

S

ee s

tud

en

ts’

wo

rk.

'

''

'

''

'

''


NA

ME

DA

TE

PE

RIO

D

Lesson 7-7

Ch

ap

ter

7

43

G

len

co

e G

eo

me

try

7-7

Sca

le M

od

els

A

sca

le m

od

el

or

a s

ca

le d

ra

win

g i

s an

obje

ct o

r d

raw

ing w

ith

len

gth

s p

rop

ort

ion

al

to t

he o

bje

ct i

t re

pre

sen

ts.

Th

e s

ca

le o

f a m

od

el

or

dra

win

g i

s th

e r

ati

o o

f th

e

len

gth

of

the m

od

el

or

dra

win

g t

o t

he a

ctu

al

len

gth

of

the o

bje

ct b

ein

g m

od

ele

d o

r d

raw

n.

MA

PS

Th

e s

ca

le o

n t

he

ma

p s

ho

wn

is 0

.75

in

ch

es :

6 m

ile

s.

Fin

d t

he

actu

al

dis

tan

ce

fro

m P

ine

ha

m t

o M

en

lo F

ield

s.

Use

a r

ule

r. T

he d

ista

nce

betw

een

Pin

eh

am

an

d

Men

lo F

ield

s is

abou

t 1.2

5 i

nch

es.

Me

tho

d 1

: W

rit

e a

nd

so

lve

a p

ro

po

rti

on

.

Let

x r

ep

rese

nt

the d

ista

nce

betw

een

cit

ies.

0.7

5 i

n.

−

6 m

i

= 1

.25 i

n.

−

x m

i

0.7

5 ·

x =

6 ·

1.2

5

Cro

ss P

rod

uct

s P

rop

ert

y

x =

10

Sim

pli

fy.

Me

tho

d 2

: W

rit

e a

nd

so

lve

an

eq

ua

tio

n.

Let

a =

act

ual

dis

tan

ce a

nd

m =

map

dis

tan

ce

in i

nch

es.

Wri

te t

he s

cale

as

6 m

i −

0.7

5 i

n. ,

wh

ich

is

6 ÷

0.7

5 o

r 8 m

iles

per

inch

.

a =

8 ·

m

Wri

te a

n e

qu

ati

on

.

= 8

· 1

.25

m =

1.2

5 i

n.

= 1

0

Solv

e.

Th

e d

ista

nce

betw

een

Pin

eh

am

an

d M

en

lo F

ield

s is

10 m

iles.

Exam

ple

ma

p

act

ual

Exerc

ises

Use

th

e m

ap

ab

ov

e a

nd

a c

usto

ma

ry

ru

ler t

o f

ind

th

e a

ctu

al

dis

tan

ce

be

twe

en

ea

ch

pa

ir o

f cit

ies.

Me

asu

re

to

th

e n

ea

re

st

six

tee

nth

of

an

in

ch

.

1

. E

ast

wic

h a

nd

Need

ham

Beach

13 m

iles

2

. N

ort

h P

ark

an

d M

en

lo F

ield

s 18 m

iles

3

. N

ort

h P

ark

an

d E

ast

wic

h 8 m

iles

4

. D

en

vil

le a

nd

Pin

eh

am

4 m

iles

5

. P

ineh

am

an

d E

ast

wic

h 13 m

iles

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n

Scale

Dra

win

gs a

nd

Mo

dels

Denvill

e Pin

eham

Eastw

ich

Nort

h P

ark

Needham

B

each

Menlo

Fie

lds

0.7

5 in.

6 m

i.

Answers (Lesson 7-6 and Lesson 7-7)


Answers

Co

pyri

gh

t ©

Gle

nco

e/M

cG

raw

-Hill, a

div

isio

n o

f T

he

Mc

Gra

w-H

ill C

om

pa

nie

s,

Inc

.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

44

G

len

co

e G

eo

me

try

Stu

dy

Gu

ide a

nd

In

terv

en

tio

n (c

on

tin

ued

)

Scale

Dra

win

gs a

nd

Mo

dels

7-7

Use

Sca

le F

acto

rs

Th

e s

ca

le f

acto

r o

f a d

raw

ing o

r sc

ale

mod

el

is t

he s

cale

wri

tten

as

a u

nit

less

rati

o i

n s

imp

lest

form

. S

cale

fact

ors

are

alw

ays

wri

tten

so t

hat

the m

od

el

len

gth

in

th

e r

ati

o c

om

es

firs

t.

SC

ALE

MO

DE

L A

do

ll h

ou

se

th

at

is 1

5 i

nch

es t

all

is a

sca

le m

od

el

of

a r

ea

l h

ou

se

wit

h a

he

igh

t o

f 2

0 f

ee

t.

a.

Wh

at

is t

he

sca

le o

f th

e m

od

el?

To f

ind

th

e s

cale

, w

rite

th

e r

ati

o o

f a m

od

el

len

gth

to a

n a

ctu

al

len

gth

.

m

od

el

len

gth

−

act

ual

len

gth

=

15 i

n.

−

20 f

t or

3 i

n.

−

4 f

t

Th

e s

cale

of

the m

od

el

is 3

in

.:4 f

t

b.

Ho

w m

an

y t

ime

s a

s t

all

as t

he

actu

al

ho

use

is t

he

mo

de

l?

Mu

ltip

ly t

he s

cale

fact

or

of

the m

od

el

by a

con

vers

ion

fact

or

that

rela

tes

inch

es

to f

eet

to

obta

in a

un

itle

ss r

ati

o.

3 i

n.

−

4 f

t =

3 i

n.

−

4 f

t ·

1 f

t

−

12 i

n. =

3

−

48 o

r 1

−

16

Th

e s

cale

fact

or

is 1

:16.

Th

at

is,

the m

od

el

is

1

−

16 a

s ta

ll a

s th

e a

ctu

al

hou

se.

Exerc

ises

1

. M

OD

EL T

RA

IN T

he l

en

gth

of

a m

od

el

train

is

18 i

nch

es.

It

is a

sca

le m

od

el

of

a t

rain

that

is 4

8 f

eet

lon

g.

Fin

d t

he s

cale

fact

or.

1:3

2

2

. A

RT

A

n a

rtis

t in

Port

lan

d,

Ore

gon

, m

ak

es

bro

nze s

culp

ture

s of

dogs.

Th

e r

ati

o o

f th

e

heig

ht

of

a s

culp

ture

to t

he a

ctu

al

heig

ht

of

the d

og i

s 2:3

. If

th

e h

eig

ht

of

the s

culp

ture

is

14 i

nch

es,

fin

d t

he h

eig

ht

of

the d

og.

21 i

n.

3

. B

RID

GE

S T

he s

pan

of

the B

en

jam

in F

ran

kli

n s

usp

en

sion

bri

dge i

n P

hil

ad

elp

hia

, P

en

nsy

lvan

ia,

is 1

750 f

eet.

A m

od

el

of

the b

rid

ge h

as

a s

pan

of

42 i

nch

es.

Wh

at

is t

he

rati

o o

f th

e s

pan

of

the m

od

el

to t

he s

pan

of

the a

ctu

al

Ben

jam

in F

ran

kli

n B

rid

ge?

1− 5

00

Exam

ple


NA

ME

DA

TE

PE

RIO

D

Lesson 7-7

Ch

ap

ter

7

45

G

len

co

e G

eo

me

try

7-7

MA

PS

U

se

th

e m

ap

sh

ow

n a

nd

a

cu

sto

ma

ry

ru

ler t

o f

ind

th

e a

ctu

al

dis

tan

ce

be

twe

en

ea

ch

pa

ir o

f cit

ies.

Me

asu

re

to

th

e n

ea

re

st

six

tee

nth

of

an

in

ch

.

1

. P

ort

Jaco

b a

nd

Sou

thp

ort

2

. P

ort

Jaco

b a

nd

Bri

gh

ton

Beach

3

. B

righ

ton

Beach

an

d P

irate

s’ C

ove

4

. E

ast

port

an

d S

an

d D

oll

ar

Reef

5

. S

CA

LE

MO

DE

L S

an

jay i

s m

ak

ing a

139 c

en

tim

ete

rs l

on

g s

cale

mod

el

of

the P

art

hen

on

for

his

Worl

d H

isto

ry c

lass

. T

he a

ctu

al

len

gth

of

the P

art

hen

on

is

69.5

mete

rs l

on

g.

a.

Wh

at

is t

he s

cale

of

the m

od

el?

b.

How

man

y t

imes

as

lon

g a

s th

e a

ctu

al

Part

hen

on

is

the m

od

el?

6

. A

RC

HIT

EC

TU

RE

A

n a

rch

itect

is

mak

ing a

sca

le m

od

el

of

an

off

ice b

uil

din

g h

e w

ish

es

to

con

stru

ct.

Th

e m

od

el

is 9

in

ches

tall

. T

he a

ctu

al

off

ice b

uil

din

g h

e p

lan

s to

con

stru

ct w

ill

be 7

5 f

eet

tall

.

a.

Wh

at

is t

he s

cale

of

the m

od

el?

b.

Wh

at

scale

fact

or

did

th

e a

rch

itect

use

to b

uil

d h

is m

od

el?

7. W

HIT

E H

OU

SE

C

raig

is

mak

ing a

sca

le d

raw

ing o

f th

e W

hit

e H

ou

se o

n a

n

8.5

-by-1

1-i

nch

sh

eet

of

pap

er.

Th

e W

hit

e H

ou

se i

s 168 f

eet

lon

g a

nd

152 f

eet

wid

e.

Ch

oose

an

ap

pro

pri

ate

sca

le f

or

the d

raw

ing a

nd

use

th

at

scale

to d

ete

rmin

e t

he

dra

win

g’s

dim

en

sion

s.

8

. G

EO

GR

AP

HY

C

hoose

an

ap

pro

pri

ate

sca

le a

nd

con

stru

ct a

sca

le d

raw

ing o

f each

rect

an

gu

lar

state

to f

it o

n a

3-b

y-5

-in

ch i

nd

ex c

ard

.

a. T

he s

tate

of

Colo

rad

o i

s ap

pro

xim

ate

ly 3

80 m

iles

lon

g (

east

to w

est

) an

d 2

80 m

iles

wid

e (

nort

h t

o s

ou

th).

b.

Th

e s

tate

of

Wyom

ing i

s ap

pro

xim

ate

ly 3

65 m

iles

lon

g (

east

to w

est

) an

d 2

65 m

iles

wid

e (

nort

h t

o s

ou

th).

Sk

ills

Pra

ctic

e

Scale

Dra

win

gs a

nd

Mo

dels

52 1

−

2 m

i

72 1

−

2 m

i

50 m

i

77 1

−

2 m

i

Port

Jacob Eastp

ort

Brighto

n B

each

Pirate

s’ C

ove

South

port

Sand D

olla

r R

eef

0.5

in.

20 m

i

1

−

500

2 c

m:1

m

3 i

n.:

25 f

t

1:1

00

1 i

n.:

18 f

t; 8

4

−

9 i

n.

× 9

1

−

3 i

n.

1 i

n.:

80 m

i; d

raw

ing

sh

ou

ld b

e a

bo

ut

4.7

5 i

n.

× 3

.5 i

n.

1 i

n.:

75 m

i; d

raw

ing

sh

ou

ld b

e a

bo

ut

4.8

7 i

n.

× 3

.53 i

n.



Co

pyrig

ht ©

Gle

nc

oe

/Mc

Gra

w-H

ill, a d

ivis

ion

of T

he

Mc

Gra

w-H

ill Co

mp

an

ies, In

c.


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

46

G

len

co

e G

eo

me

try

Practi

ce

Scale

Dra

win

gs a

nd

Mo

dels

7-7

MA

PS

U

se

th

e m

ap

of

Ce

ntr

al

Ne

w J

erse

y s

ho

wn

an

d a

n i

nch

ru

ler t

o f

ind

th

e

actu

al

dis

tan

ce

be

twe

en

ea

ch

pa

ir o

f cit

ies.

Me

asu

re

to

th

e n

ea

re

st

six

tee

nth

of

an

in

ch

.

1

. H

igh

lan

d P

ark

an

d M

etu

chen

ab

ou

t 4.4

65 m

iles

2

. N

ew

Bru

nsw

ick

an

d R

obin

vale

ab

ou

t 6.1

1 m

iles

3

. R

utg

ers

Un

ivers

ity L

ivin

gst

on

Cam

pu

s an

d R

utg

ers

Un

ivers

ity

Cook

–D

ou

gla

ss C

am

pu

s

ab

ou

t 3.4

075 m

iles

4

. A

IRP

LA

NE

S W

illi

am

is

bu

ild

ing a

scale

mod

el

of

a B

oein

g 7

47–400 a

ircr

aft

.

Th

e w

ingsp

an

of

the m

od

el

is a

pp

roxim

ate

ly 8

feet

10

1 − 16

in

ches.

If

the s

cale

fact

or

of

the

mod

el

is a

pp

roxim

ate

ly 1

:24,

wh

at

is t

he a

ctu

al

win

gsp

an

of

a B

oein

g 7

47–400 a

ircr

aft

?

212.1

25 f

eet

5

. E

NG

INE

ER

ING

A

civ

il e

ngin

eer

is m

ak

ing a

sca

le m

od

el

of

a h

igh

way o

n r

am

p.

Th

e l

en

gth

of

the m

od

el

is 4

in

ches

lon

g.

Th

e a

ctu

al

len

gth

of

the o

n r

am

p i

s 500 f

eet.

a.

Wh

at

is t

he s

cale

of

the m

od

el?

1 i

n.:

125 f

t

b.

How

man

y t

imes

as

lon

g a

s th

e a

ctu

al

on

ram

p i

s th

e m

od

el?

1500

c.

How

man

y t

imes

as

lon

g a

s th

e m

od

el

is t

he a

ctu

al

on

ram

p?

500

6

. M

OV

IES

A

movie

dir

ect

or

is c

reati

ng a

sca

le m

od

el

of

the E

mp

ire S

tate

Bu

ild

ing t

o

use

in

a s

cen

e.

Th

e E

mp

ire S

tate

Bu

ild

ing i

s 1250 f

eet

tall

.

a.

If t

he m

od

el

is 7

5 i

nch

es

tall

, w

hat

is t

he s

cale

of

the m

od

el?

3 i

n.:

50 f

t

b.

How

tall

wou

ld t

he m

od

el

be i

f th

e d

irect

or

use

s a s

cale

fact

or

of

1:7

5?

16 f

eet

8 i

nch

es

7

. M

ON

A L

ISA

A

vis

itor

to t

he L

ou

vre

Mu

seu

m i

n P

ari

s w

an

ts t

o s

ketc

h a

dra

win

g

of

the M

on

a L

isa

, a f

am

ou

s p

ain

tin

g.

Th

e o

rigin

al

pain

tin

g i

s 77 c

en

tim

ete

rs b

y

53 c

en

tim

ete

rs.

Ch

oose

an

ap

pro

pri

ate

sca

le f

or

the r

ep

lica

so t

hat

it w

ill

fit

on

a

8.5

-by-1

1-i

nch

sh

eet

of

pap

er.

1 i

n.:

7 c

m

1 in.

1.8

8 m

i

Hig

hla

nd

Pa

rk

Me

tuch

en

Ne

wB

run

sw

ick

Ro

bin

va

le

Ru

tge

rsU

niv

-Co

ok-D

ou

gla

ss

Ca

mp

us

Ru

tge

rsU

niv

-Liv

ing

sto

nC

am

pu

s


NA

ME

DA

TE

PE

RIO

D

Lesson 7-7

Ch

ap

ter

7

47

G

len

co

e G

eo

me

try

7-7

1

. M

OD

ELS

L

uk

e w

an

ts t

o m

ak

e a

sca

le

mod

el

of

a B

oein

g 7

47 j

etl

iner.

He w

an

ts

every

foot

of

his

mod

el

to r

ep

rese

nt

50 f

eet.

Com

ple

te t

he f

oll

ow

ing t

able

.

Pa

rt

Ac

tua

l

len

gth

(in

.)

Mo

de

l

len

gth

(in

.)

Win

g S

pa

n2

53

7

50.7

4

Le

ng

th2

78

2

55.6

4

Ta

il H

eig

ht

39

2

7.8

4(S

ou

rce:

Boein

g)

2

. P

HO

TO

GR

AP

HS

T

racy

is

4 f

eet

tall

an

d

her

fath

er

is 6

feet

tall

. In

a p

hoto

gra

ph

of

the t

wo o

f th

em

sta

nd

ing s

ide b

y s

ide,

Tra

cy’s

im

age i

s 2 i

nch

es

tall

.

6 ft

4 ft

Alt

hou

gh

th

eir

im

ages

are

mu

ch

small

er,

th

e r

ati

o o

f th

eir

heig

hts

rem

ain

s th

e s

am

e.

How

tall

is

Tra

cy’s

fath

er’

s im

age i

n t

he p

hoto

? W

hat

is

the s

cale

of

the p

hoto

?

1:2

4

3

. T

OW

ER

S T

he T

ok

yo T

ow

er

in J

ap

an

is

curr

en

tly t

he w

orl

d’s

tall

est

self

-

sup

port

ing s

teel

tow

er.

It

is 3

33 m

ete

rs

tall

.

a.

Heero

bu

ild

s a m

od

el

of

the T

ok

yo

Tow

er

that

is 2

775 m

illi

mete

rs t

all

.

Wh

at

is t

he s

cale

of

Heero

’s m

od

el?

25 m

m:3

m

b.

How

man

y t

imes

as

tall

as

the a

ctu

al

tow

er

is t

he m

od

el?

4

. P

UP

PIE

S M

ere

dit

h’s

new

Pom

era

nia

n

pu

pp

y i

s 7 i

nch

es

tall

an

d 9

in

ches

lon

g.

Sh

e w

an

ts t

o m

ak

e a

dra

win

g o

f h

er

new

Pom

era

nia

n t

o p

ut

in h

er

lock

er.

If

the

sheet

of

pap

er

she i

s u

sin

g i

s 3 i

nch

es

by

5 i

nch

es,

fin

d a

n a

pp

rop

riate

sca

le f

act

or

for

Mere

dit

h t

o u

se i

n h

er

dra

win

g.

1:2

.5 o

r 2:5

(an

sw

ers

may v

ary

, b

ut

sh

ou

ld b

e a

t le

ast

gre

ate

r th

an

1:2

.34)

5

. M

AP

SC

arl

os

mak

es

a m

ap

of

his

neig

hborh

ood

for

a p

rese

nta

tion

. T

he

scale

of

his

map

is

1 i

nch

:125 f

eet.

a.

How

man

y f

eet

do 4

in

ches

rep

rese

nt

on

th

e m

ap

?

500 f

t

b.

Carl

os

lives

250 f

eet

aw

ay f

rom

An

dre

w.

How

man

y i

nch

es

sep

ara

te

Carl

os’

hom

e f

rom

An

dre

w’s

on

the m

ap

?2 i

n.

c.

Du

rin

g a

pra

ctic

e r

un

in

fro

nt

of

his

pare

nts

, C

arl

os

reali

zes

that

his

map

is f

ar

too s

mall

. H

e d

eci

des

to m

ak

e

his

map

5 t

imes

as

larg

e.

Wh

at

wou

ld

be t

he s

cale

of

the l

arg

er

map

?

1 i

n.:

25 f

t

Wo

rd

Pro

ble

m P

racti

ce

Scale

Dra

win

gs a

nd

Mo

dels

1

−

12

0



Answers

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NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

7

48

G

len

co

e G

eo

me

try

7-7

Are

a a

nd

Vo

lum

e o

f S

cale

Mo

dels

an

d D

raw

ing

s

You

have a

lread

y l

earn

ed

abou

t ch

an

ges

of

len

gth

measu

res

betw

een

sca

le m

od

els

an

d t

he

obje

ct t

hat

is b

ein

g m

od

ele

d.

Th

e a

reas

an

d v

olu

mes

of

scale

mod

els

an

d d

raw

ings

als

o

chan

ge,

bu

t by m

ult

iple

s d

iffe

ren

t th

an

th

e “

scale

fact

or.

”

Y

ua

n i

s m

ak

ing

a s

ca

le m

od

el

of

a c

yli

nd

er.

Th

e a

ctu

al

cy

lin

de

r h

as

a r

ad

ius o

f r i

nch

es a

nd

a h

eig

ht

of h

in

ch

es.

Th

e s

ca

le f

acto

r o

f th

e m

od

el

is 1

:2.

Wh

at

is t

he

ra

tio

of

vo

lum

es o

f th

e m

od

el

cy

lin

de

r t

o t

he

actu

al

cy

lin

de

r?

Th

e v

olu

me f

orm

ula

for

a c

yli

nd

er

is V

= π

r2h

. T

he a

ctu

al

cyli

nd

er’

s volu

me i

s π

r2h

.

If t

he c

yli

nd

er

is s

cale

d d

ow

n b

y a

fact

or

of

1:2

, th

e n

ew

rad

ius

wil

l be 1

−

2 r

an

d t

he n

ew

heig

ht

wil

l be 1

−

2 h

.

V =

π ( 1

−

2 r

) 2

( 1

−

2 h

)

= 1

−

8 π

r2h

Th

e m

od

el’s

volu

me i

s 1

−

8 o

f th

e a

ctu

al

volu

me.

Exerc

ises

1

. C

on

sid

er

a p

ain

tin

g o

n a

22-i

nch

-by-2

8-i

nch

can

vas.

Su

pp

ose

you

wis

h t

o m

ak

e a

1:2

sc

ale

mod

el

of

the p

ain

tin

g f

or

art

cla

ss.

Wh

at

is t

he r

ati

o o

f are

as

of

the m

od

el

pain

tin

g

to t

he a

ctu

al

pain

tin

g?

(Are

a =

Len

gth

× W

idth

) 1

−

4

2

. T

he P

ark

s an

d R

ecr

eati

on

Off

ice i

s p

lan

nin

g a

new

cir

cula

r p

laygro

un

d w

ith

a r

ad

ius

of

30 f

eet.

Befo

re t

hey c

an

con

stru

ct t

he p

laygro

un

d,

they a

sk a

n a

rch

itect

to c

reate

a 1

:20

scale

mod

el

of

the p

rop

ose

d p

laygro

un

d s

uch

th

at

the n

ew

rad

ius

is 1

.5 f

eet.

Wh

at

is t

he

rati

o o

f are

as

of

the m

od

el

pla

ygro

un

d t

o t

he p

rop

ose

d a

ctu

al

pla

ygro

un

d?

(Are

a =

π ×

(ra

diu

s)2)

1

−

40

0

3

. A

refr

igera

tor

man

ufa

ctu

rer

use

s a 7

-foot-

by-3

-foot-

by-3

-foot

box f

or

its

stan

dard

mod

el.

T

he m

ark

eti

ng t

eam

su

ggest

s th

e m

an

ufa

ctu

rer

start

sell

ing a

sm

all

er,

low

er-

pri

ced

re

frig

era

tor

wit

h a

sca

le f

act

or

of

4:5

to t

he s

tan

dard

mod

el.

If

the b

ox i

s re

du

ced

by a

si

mil

ar

scale

, w

hat

is t

he r

ati

o o

f volu

mes

of

the n

ew

, sm

all

er

box t

o t

he c

urr

en

t box?

(Volu

me =

len

gth

× w

idth

× h

eig

ht)

6

4

−

12

5

4

. C

on

sid

er

a c

ube w

ith

sid

e l

en

gth

s x.

If e

ach

sid

e o

f th

e c

ube i

s sc

ale

d b

y a

fact

or

of

1:y

, w

hat

is t

he r

ati

o o

f volu

mes

of

the m

od

el

cube t

o t

he a

ctu

al

cube?

(Volu

me =

len

gth

× w

idth

× h

eig

ht)

( 1

−

y ) 3

En

ric

hm

en

t

Exam

ple



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Chapter 7 Assessment Answer KeyQuiz 1 (Lessons 7-1 and 7-2) Quiz 3 (Lessons 7-5 and 7-6) Mid-Chapter Test

Page 51 Page 52 Page 53

Quiz 2 (Lessons 7-3 and 7- 4) Quiz 4 (Lesson 7-7)

Page 51 Page 52

1.

2.

3.

4.

5.

20, 65, 95

276 students

△BAC ∼ △QPR or △ABC ∼ △QPR by

SAS.

x = 4; scale factor 5:4

B

1.

2.

3.

4.

5.

yes; AA

No; the sides are

not proportional.

△ABD ∼ △CDE

or △ADB ∼ △CDF; 8

Not parallel because

JM −

MK ≠ JN

− NL

20

1.

2.

3.

4.

5.

2.3 cm

corresp. altitudes are

proportional to the

corresp. sides

5.3 cm

reduction; 1 −

2

y

x

1.

2.

3.

4.

2:5

33 in.

1:3.5

D

1.

2.

3.

4.

5.

B

J

C

J

B

6.

7.

8.

9.

10.

No; SAS does not apply.

yes; SAS

Not parallel because

JM

− MK

≠ JN

− LN

9

2.9 in.

Answers

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Chapter 7 Assessment Answer KeyVocabulary Test Form 1

Page 54 Page 55 Page 56

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

ratio

scale factor


extremes

means

false; similar polygons

false; proportion

true

Sample answer: product of means is equal to product of

extremes

Sample answer: a segment with

endpoints that are the midpoints of any two

sides of a triangle

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

A

J

B

G

B

J

B

H

A

G

11.

12.

13.

14.

15.

16.

17.

18.

19.

D

J

B

F

B

G

D

H

D

B: 9.6

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Chapter 7 Assessment Answer KeyForm 2A Form 2B

Page 57 Page 58 Page 59 Page 60

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

A

F

D

H

B

H

C

F

C

G

11.

12.

13.

14.

15.

16.

17.

18.

D

J

B

G

C

F

A

J

B: 32

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

A

H

B

J

B

G

D

G

B

H

11.

12.

13.

14.

15.

16.

17.

18.

D

F

C

J

B

F

D

H

B: 26

Answers

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Chapter 7 Assessment Answer KeyForm 2C

Page 61 Page 62

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

3:7

Yes; corres. are !.

30 ft

18

42.5 ft

6.3

18.2

72

△PQR ∼ △STR

2 1 − 2

Yes. The right angles

are congruent. The

legs are proportional.

By SAS similarity, the


12.

13.

14.

15.

16.

17.

18.

B:

9.5

42

47

2

3

14.4

4

7

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Chapter 7 Assessment Answer KeyForm 2D

Page 63 Page 64

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

5:11

No; the corresponding sides are not proportional.

48 ft

4 −

3

188 1 −

3 ft

5.5

15

75

△XYZ ∼ △XNM

7.5

No. The legs of the

right triangles are

not proportional.

12.

13.

14.

15.

16.

17.

18.

B:

2.2

96

85

9.5

2

5 5 −

8

28

4.8

Answers

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Chapter 7 Assessment Answer KeyForm 3

Page 65 Page 66

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

4:3

No; corr. sides are

not proportional.

yes; SAS

30 cm by 7.5 cm

5.85

5

104 in.

82

48

8 −

3

5

12.

13.

14.

15.

16.

17.

18.

B:

11 3 −

7

7.2

8 −

5

60

24 ft

9

25.6

(0, 0), (8, 0), (0, 8)

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Chapter 7 Assessment Answer KeyExtended-Response Test, Page 67

Scoring Rubric

Score General Description Speci! c Criteria

4 Superior

A correct solution that

is supported by well-developed,

accurate explanations

• Shows thorough understanding of the concepts of ratios,

properties of proportions, similar ! gures, similar triangles,

dividing segments into parts, proportional parts of triangles,

corresponding perimeters and altitudes.

• Uses appropriate strategies to solve problems.

• Computations are correct.

• Written explanations are exemplary.

• Figures are accurate and appropriate.

• Goes beyond requirements of some or all problems.

3 Satisfactory

A generally correct solution, but may

contain minor ! aws in reasoning or

computation

• Shows an understanding of the concepts of ratios,


dividing segments into parts, proportional parts of

triangles, corresponding perimeters and altitudes.

• Uses appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are effective.

• Figures are mostly accurate and appropriate.

• Satis" es all requirements of problems.

2 Nearly Satisfactory

A partially correct interpretation and/or

solution to the problem

• Shows an understanding of most of the concepts of ratios,


dividing segments into parts, proportional parts of triangles,

corresponding perimeters and altitudes.

• May not use appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are satisfactory.

• Figures are mostly accurate.

• Satis" es the requirements of most of the problems.

1 Nearly Unsatisfactory

A correct solution with no supporting

evidence or explanation

• Final computation is correct.

• No written explanations or work is shown to substantiate the

" nal computation.

• Figures may be accurate but lack detail or explanation.

• Satis" es minimal requirements of some of the problems.

0 Unsatisfactory

An incorrect solution indicating no

mathematical understanding of the

concept or task, or no solution is given

• Shows little or no understanding of most of the concepts of

ratios, properties of proportions, similar ! gures, similar

triangles, dividing segments into parts, proportional parts

of triangles, corresponding perimeters and altitudes.

• Does not use appropriate strategies to solve problems.

• Computations are incorrect.

• Written explanations are unsatisfactory.

• Figures are inaccurate or inappropriate.

• Does not satisfy requirements of problems.

• No answer may be given.

Answers

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Chapter 7 Assessment Answer KeyExtended-Response Test, Page 67

Sample Answers

In addition to the scoring rubric found on page A30, the following sample answers may be used as guidance in evaluating open-ended assessment items.

1. 10

− 6 =

5 −

3 or

10 −

2 =

15 −

3

2a. 3:2:2 (Note: Check to be sure that the sum of any two sides of the triangle is greater than the third side. For example, a ratio of 1:1:2 would not be acceptable.)

2b. 18, 12, 12

2c. 12, 8, 8

3. △ABH ∼ △CDI ∼ △GFI ∼ △ADG ∼ △GDE ∼ △CFE ∼ △AGE by the AA Postulate.

4.

FD

CA

B

1

1

E

4–3

3–4

5. Mark the midpoint of each side of △PQR. Connect these points to form △XYZ. Since each segment of △XYZ is a midsegment of △PQR, its length will be half the corresponding side, and therefore the perimeter will be

half the perimeter of △PQR.

P R

Q

X

Y Z

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Chapter 7 Assessment Answer KeyStandardized Test Practice

Page 68 Page 69

1.

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

A B C D

8. F G H J

9. A B C D

10. F G H J

11. A B C D

12. F G H J

13.

14.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

1 2 8

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

1 6

Answers

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Chapter 7 Assessment Answer KeyStandardized Test Practice (continued)

Page 70

15.

16.

17.

18.

19.

20

21a.

21b.

21c.

-3

85.5

11 < x < 53

64.5 cm

congruent; TU = WX = √ "" 40 UV = XY = √ "" 53 VT = YU = √ "" 41

m∠1 = 86.5,

m∠2 = 93.5

ST = 6.32, PR = 28.46

slope of −− ST = 1 −

3 ,

slope of −− PR = -3

perpendicular

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