chapter 9 resource masters - math class · 2019-11-24 · ©glencoe/mcgraw-hill iv glencoe geometry...

Chapter 9Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 9 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. 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’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860186-X GeometryChapter 9 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix

Lesson 9-1Study Guide and Intervention . . . . . . . . 479–480Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 481Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 482Reading to Learn Mathematics . . . . . . . . . . 483Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 484







Chapter 9 AssessmentChapter 9 Test, Form 1 . . . . . . . . . . . . 521–522Chapter 9 Test, Form 2A . . . . . . . . . . . 523–524Chapter 9 Test, Form 2B . . . . . . . . . . . 525–526Chapter 9 Test, Form 2C . . . . . . . . . . . 527–528Chapter 9 Test, Form 2D . . . . . . . . . . . 529–530Chapter 9 Test, Form 3 . . . . . . . . . . . . 531–532Chapter 9 Open-Ended Assessment . . . . . . 533Chapter 9 Vocabulary Test/Review . . . . . . . 534Chapter 9 Quizzes 1 & 2 . . . . . . . . . . . . . . . 535Chapter 9 Quizzes 3 & 4 . . . . . . . . . . . . . . . 536Chapter 9 Mid-Chapter Test . . . . . . . . . . . . 537Chapter 9 Cumulative Review . . . . . . . . . . . 538Chapter 9 Standardized Test Practice . 539–540

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 9 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 9 Resource Masters includes the core materials neededfor Chapter 9. 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 in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.

WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

© Glencoe/McGraw-Hill v Glencoe Geometry

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

Assessment OptionsThe assessment masters in the Chapter 9Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

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

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 518–519. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

79

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 9.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

component form

dilation

isometry

line of reflection

line of symmetry

point of symmetry

reflection

regular tessellation

resultant

rotation

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

rotational symmetry

scalar

semi-regular tessellation

similarity transformation

standard position

tessellation

transformation

translation

uniform tessellations

vector

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

99

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

99

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 9. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 9.1

Theorem 9.2

Postulate 9.1

Study Guide and InterventionReflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

© Glencoe/McGraw-Hill 479 Glencoe Geometry

Less

on

9-1

Draw Reflections The transformation called a reflection is a flip of a figure in a point, a line, or a plane. The new figure is the image and the original figure is thepreimage. The preimage and image are congruent, so a reflection is a congruencetransformation or isometry.

Construct the imageof quadrilateral ABCD under areflection in line m .

Draw a perpendicular from each vertexof the quadrilateral to m. Find verticesA!, B!, C!, and D! that are the samedistance from m on the other side of m.The image is A!B!C!D!.

m

C AD

B

A!

B!

C! D!

Quadrilateral DEFG hasvertices D("2, 3), E(4, 4), F(3, "2), and G("3, "1). Find the image under reflectionin the x-axis.

To find an image for areflection in the x-axis,use the same x-coordinateand multiply the y-coordinate by "1. Insymbols, (a, b) → (a, "b).The new coordinates areD!("2, "3), E!(4, "4),F!(3, 2), and G!("3, 1).The image is D!E!F !G!.

x

y

O

D!E!

F!G!

DE

FG

Example 1Example 1 Example 2Example 2

In Example 2, the notation (a, b) → (a, "b) represents a reflection in the x-axis. Here arethree other common reflections in the coordinate plane.• in the y-axis: (a, b) → ("a, b)• in the line y # x: (a, b) → (b, a)• in the origin: (a, b) → ("a, "b)

Draw the image of each figure under a reflection in line m .

1. 2. 3.

Graph each figure and its image under the given reflection.

4. !DEF with D("2, "1), E("1, 3), 5. ABCD with A(1, 4), B(3, 2), C(2, "2),F(3, "1) in the x-axis D("3, 1) in the y-axis

x

y

OD!

C!

B!

A!

D

C

B

A

x

y

O

F!D!

E!

FD

E

m

QR S

TUm

LM

N

OP

m

HJ

K

ExercisesExercises


Lines and Points of Symmetry If a figure has a line of symmetry, then it can befolded along that line so that the two halves match. If a figure has a point of symmetry, itis the midpoint of all segments between the preimage and image points.

Determine how many lines of symmetry a regular hexagon has. Then determine whether a regular hexagon has point symmetry.There are six lines of symmetry, three that are diagonals through opposite vertices and three that are perpendicular bisectors of opposite sides. The hexagon has point symmetrybecause any line through P identifies two points on the hexagon that can be considered images of each other.

Determine how many lines of symmetry each figure has. Then determine whetherthe figure has point symmetry.

1. 2. 3.

4. 5. 6.

7. 8. 9.

A B

C

DE

FP

Study Guide and Intervention (continued)

Reflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

ExampleExample

ExercisesExercises

Skills PracticeReflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1


Less

on

9-1

Draw the image of each figure under a reflection in line !.

1. 2.

COORDINATE GEOMETRY Graph each figure and its image under the givenreflection.

3. !ABC with vertices A("3, 2), B(0, 1), 4. trapezoid DEFG with vertices D(0, "3),and C("2, "3) in the origin E(1, 3), F(3, 3), and G(4, "3) in the y-axis

5. parallelogram RSTU with vertices 6. square KLMN with vertices K("1, 0),R("2, 3), S(2, 4), T(2, "3) and L("2, 3), M(1, 4), and N(2, 1) in U("2, "4) in the line y # x the x-axis

Determine how many lines of symmetry each figure has. Then determine whetherthe figure has point symmetry.

7. 8. 9.

M!

K!K

x

y

O

L!

N!

LM

Nx

y

O

R!

S!T!

U!

RS

TU

x

y

O

D

E F

GD!

E!F!

G!

x

y

O

A!B!

C!A

B

C

!

!


Draw the image of each figure under a reflection in line !.

1. 2.

COORDINATE GEOMETRY Graph each figure and its image under the givenreflection.

3. quadrilateral ABCD with vertices 4. !FGH with vertices F("3, "1), G(0, 4),A("3, 3), B(1, 4), C(4, 0), and and H(3, "1) in the line y # xD("3, "3) in the origin

5. rectangle QRST with vertices Q("3, 2), 6. trapezoid HIJK with vertices H("2, 5),R("1, 4), S(2, 1), and T(0, "1) I(2, 5), J("4, "1), and K("4, 3) in the x-axis in the y-axis

ROAD SIGNS Determine how many lines of symmetry each sign has. Thendetermine whether the sign has point symmetry.

7. 8. 9.

H I

J

K

H!I!

J!

K!

x

y

OQ!

R!

S!

T!Q

R

S

Tx

y

O

F!

G!

H!

F

G

Hx

y

O

A!B!

C!

D!AB

C

D

x

y

O

!!

Practice Reflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

Reading to Learn MathematicsReflections

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1


Less

on

9-1

Pre-Activity Where are reflections found in nature?

Read the introduction to Lesson 9-1 at the top of page 463 in your textbook.

Suppose you draw a line segment connecting a point at the peak of a mountainto its image in the lake. Where will the midpoint of this segment fall?

Reading the Lesson1. Draw the reflected image for each reflection described below.

a. reflection of trapezoid ABCD in the line n b. reflection of !RST in point PLabel the image of ABCD as A!B!C!D!. Label the image of RST as R!S!T!.

c. reflection of pentagon ABCDE in the originLabel the image of ABCDE as A!B!C!D!E!.

2. Determine the image of the given point under the indicated reflection.a. (4, 6); reflection in the y-axisb. ("3, 5); reflection in the x-axisc. ("8, "2); reflection in the line y # xd. (9, "3); reflection in the origin

3. Determine the number of lines of symmetry for each figure described below. Thendetermine whether the figure has point symmetry and indicate this by writing yes or no.a. a square b. an isosceles triangle (not equilateral) c. a regular hexagon d. an isosceles trapezoid e. a rectangle (not a square) f. the letter E

Helping You Remember4. A good way to remember a new geometric term is to relate the word or its parts to

geometric terms you already know. Look up the origins of the two parts of the wordisometry in your dictionary. Explain the meaning of each part and give a term youalready know that shares the origin of that part.

A!B!

C! D!E!

AB

CDE

x

y

O

R!S!

T!

RS P

Tn

A!B!

C!D!

A

BCD


Reflections in the Coordinate Plane

Study the diagram at the right. It shows how the triangle ABC is mapped onto triangle XYZ by thetransformation (x, y) → ("x # 6, y). Notice that !XYZis the reflection image with respect to the vertical line with equation x $ 3.

1. Prove that the vertical line with equation x # 3 is theperpendicular bisector of the segment with endpoints (x, y) and ("x $ 6, y). (Hint: Use the midpoint formula.)

2. Every transformation of the form (x, y) → ("x $ 2h, y) is a reflection with respect to the vertical line with equation x # h. What kind of transformation is (x, y) → ( x, "y $ 2 k)?

Draw the transformation image for each figure and the given transformation. Is it a reflection transformation? If so, with respect to what line?

3. (x, y) → ("x " 4, y) 4. (x, y) → (x, "y $ 8)

x

y

Ox

y

O

x $ 3

A

B

C

X

Y

Z

(x, y) → ("x # 6, y)

x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

Study Guide and InterventionTranslations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

Translations Using Coordinates A transformation called a translation slides afigure in a given direction. In the coordinate plane, a translation moves every preimagepoint P(x, y) to an image point P(x $ a, y $ b) for fixed values a and b. In words, atranslation shifts a figure a units horizontally and b units vertically; in symbols,(x, y) → (x $ a, y $ b).

Rectangle RECT has vertices R("2, "1),E("2, 2), C(3, 2), and T(3, "1). Graph RECT and its image for the translation (x, y) → (x # 2, y " 1).The translation moves every point of the preimage right 2 units and down 1 unit.(x, y) → (x $ 2, y " 1)R("2, "1) → R!("2 $ 2, "1 " 1) or R!(0, "2)E("2, 2) → E!("2 $ 2, 2 " 1) or E!(0, 1)C(3, 2) → C!(3 $ 2, 2 " 1) or C!(5, 1)T(3, "1) → T!(3 $ 2, "1 " 1) or T!(5, "2)

Graph each figure and its image under the given translation.

1. P!Q! with endpoints P("1, 3) and Q(2, 2) under the translation left 2 units and up 1 unit

2. !PQR with vertices P("2, "4), Q("1, 2), and R(2, 1) under the translation right 2 units and down 2 units

3. square SQUR with vertices S(0, 2), Q(3, 1), U(2, "2), and R("1, "1) under the translation right 3 units and up 1 unit

x

y

R!

S!Q!

U!R

SQ

U

x

y

P!

Q!

R!

P

QR

x

yP!Q!

PQ

x

y

O

E!

R!

C!

T!

E

R

C

T

ExampleExample

ExercisesExercises


Translations by Repeated Reflections Another way to find the image of atranslation is to reflect the figure twice in parallel lines. This kind of translation is called acomposite of reflections.

In the figure, m || n . Find the translation image of !ABC.!A!B!C! is the image of !ABC reflected in line m .!A%B%C% is the image of !A!B!C! reflected in line n .The final image, !A%B%C%, is a translation of !ABC.

In each figure, m || n . Find the translation image of each figure by reflecting it inline m and then in line n .

1. 2.

3. 4.

5. 6. m

U

A

D

Q Q %

U %

A%

D %

U!

A!

D!

n

m

n

U

T

S

R

U %T %

S %R %

m

n

D!

Q!

AQ

U

D

A%

Q %

D %

U %T!

m n

P!P P %

E!E E %N N %

T T %

A A%

m n

A!

AB

C

A%

B%

C %

m n

A!

B!

C!

A

B

C

A%

B%

C %

m n

A!

B!

C!

A

B

C

A%

B%

C %


Translations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

ExercisesExercises

ExampleExample

Skills PracticeTranslations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

In each figure, a || b. Determine whether figure 3 is a translation image of figure 1.Write yes or no. Explain your answer.

1. 2.

3. 4.

COORDINATE GEOMETRY Graph each figure and its image under the giventranslation.

5. !JKL with vertices J("4, "4), 6. quadrilateral LMNP with vertices L(4, 2),K("2, "1), and L(2, "4) under the M(4, "1), N(0, "1), and P(1, 4) under the translation (x, y) → (x $ 2, y $ 5) translation (x, y) → (x " 4, y " 3)

M!

L

MN

P

L!

N!

P!

x

y

OJ!

K!

L!

J

K

L

x

y

O

a b

12

3

a b1

23

a

b

1

2

3a b

1 2 3


In each figure, c || d . Determine whether figure 3 is a translation image of figure 1.Write yes or no. Explain your answer.

1. 2.

COORDINATE GEOMETRY Graph each figure and its image under the giventranslation.

3. quadrilateral TUWX with vertices 4. pentagon DEFGH with vertices D("1, "2),T("1, 1), U(4, 2), W(1, 5), and X("1, 3) E(2, "1), F(5, "2), G(4, "4), H(1, "4) under the translation under the translation (x, y) → (x " 2, y " 4) (x, y) → (x " 1, y $ 5)

ANIMATION Find the translation that moves the figure on the coordinate plane.

5. figure 1 → figure 2



x

y

O

1

3

4

2

D!E!

F!

G!H!

DE

F

GH

x

y

O

W

T!U!

W!

X!

T U

X

x

y

O

c

d

1

2

3

c d1

23

Practice Translations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

Reading to Learn MathematicsTranslations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

Pre-Activity How are translations used in a marching band show?


How do band directors get the marching band to maintain the shape of thefigure they originally formed?

Reading the Lesson1. Underline the correct word or phrase to form a true statement.

a. All reflections and translations are (opposites/isometries/equivalent).b. The preimage and image of a figure under a reflection in a line have

(the same orientation/opposite orientations).c. The preimage and image of a figure under a translation have

(the same orientation/opposite orientations).d. The result of successive reflections over two parallel lines is a

(reflection/rotation/translation).e. Collinearity (is/is not) preserved by translations.f. The translation (x, y) → (x $ a, y $ b) shifts every point a units

(horizontally/vertically) and y units (horizontally/vertically).

2. Find the image of each preimage under the indicated translation.a. (x, y); 5 units right and 3 units upb. (x, y); 2 units left and 4 units downc. (x, y); 1 unit left and 6 units upd. (x, y); 7 units righte. (4, "3); 3 units upf. ("5, 6); 3 units right and 2 units downg. ("7, 5); 7 units right and 5 units downh. ("9, "2); 12 units right and 6 units down

3. !RST has vertices R("3, 3), S(0, "2), and T(2, 1). Graph !RSTand its image !R!S!T! under the translation (x, y) → (x $ 3, y " 2).List the coordinates of the vertices of the image.

Helping You Remember4. A good way to remember a new mathematical term is to relate it to an everyday meaning

of the same word. How is the meaning of translation in geometry related to the idea oftranslation from one language to another?

R!

S!

T!

R

S

T

x

y

O


Translations in The Coordinate Plane

You can use algebraic descriptions of reflections to show thatthe composite of two reflections with respect to parallel lines isa translation (that is, a slide).

1. Suppose a and b are two different real numbers. Let S and T be thefollowing reflections.

S: (x, y) → ("x $ 2 a, y)T: (x, y) → ("x $ 2 b, y)

S is reflection with respect to the line with equation x # a, and T isreflection with respect to the line with equation x # b.

a. Find an algebraic description (similar to those above for S andT) to describe the composite transformation “S followed by T.”

b. Find an algebraic description for the composite transformation“T followed by S.”

2. Think about the results you obtained in Exercise 1. What do theytell you about how the distance between two parallel lines isrelated to the distance between a preimage and image point for acomposite of reflections with respect to these lines?

3. Illustrate your answers to Exercises 1 and 2 with sketches. Use aseparate sheet if necessary.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

Study Guide and InterventionRotations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3


Less

on

9-3

Draw Rotations A transformation called a rotation turns a figure through a specifiedangle about a fixed point called the center of rotation. To find the image of a rotation, oneway is to use a protractor. Another way is to reflect a figure twice, in two intersecting lines.

!ABC has vertices A(2, 1), B(3, 4), and C(5, 1). Draw the image of !ABC under a rotation of 90°counterclockwise about the origin.• First draw !ABC. Then draw a segment from O, the origin,

to point A.• Use a protractor to measure 90° counterclockwise with O!A!

as one side.• Draw OR!!".• Use a compass to copy O!A! onto OR!!". Name the segment O!A!!!.• Repeat with segments from the origin to points B and C.

Find the image of !ABC under reflection in lines m and n .First reflect !ABC in line m. Label the image !A!B!C!.Reflect !A!B!C! in line n. Label the image !A%B%C%.!A%B%C% is a rotation of !ABC. The center of rotation is theintersection of lines m and n. The angle of rotation is twice themeasure of the acute angle formed by m and n.

Draw the rotation image of each figure 90° in the given direction about the centerpoint and label the coordinates.

1. P!Q! with endpoints P("1, "2) 2. !PQR with vertices P("2, "3), Q(2, "1),and Q(1, 3) counterclockwise and R(3, 2) clockwise about the point T(1, 1)about the origin

Find the rotation image of each figure by reflecting it in line m and then in line n.

3. 4. mn

A B

C

m nQ

P

x

yP!

Q!R!

P

Q

RT

x

y

P!

Q!

P

Q

m

n

A!

B!C!

A%

B%C%

B

C

A

x

y

O

A!B!

C!

A

B

C

R

ExercisesExercises

Example 1Example 1

Example 2Example 2


Rotational Symmetry When the figure at the right is rotated about point P by 120° or 240°, the image looks like the preimage. The figure has rotational symmetry, which means it can be rotated less than 360° about a point and the preimage and image appear to be the same.

The figure has rotational symmetry of order 3 because there are 3 rotations less than 360° (0°, 120°, 240°) that produce an image that is the same as the original. The magnitude of the rotational symmetry for a figure is 360 degrees divided by the order. For the figure above,the rotational symmetry has magnitude 120 degrees.

Identify the order and magnitude of the rotational symmetry of the design at the right.

The design has rotational symmetry about the center point for rotations of 0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°.

There are eight rotations less than 360 degrees, so the order of its rotational symmetry is 8. The quotient 360 & 8 is 45, so the magnitude of its rotational symmetry is 45 degrees.

Identify the order and magnitude of the rotational symmetry of each figure.

1. a square 2. a regular 40-gon

3. 4.

5. 6.

P


Rotations

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ExercisesExercises

ExampleExample

Skills PracticeRotations

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Rotate each figure about point R under the given angle of rotation and the givendirection. Label the vertices of the rotation image.

1. 90° counterclockwise 2. 90° clockwise

COORDINATE GEOMETRY Draw the rotation image of each figure 90° in the givendirection about the origin and label the coordinates.

3. !STW with vertices S(2, "1), T(5, 1), 4. !DEF with vertices D("4, 3), E(1, 2),and W(3, 3) counterclockwise and F("3, "3) clockwise

Use a composition of reflections to find the rotation image with respect to lines kand m. Then find the angle of rotation for each image.

5. 6. k

mL!

M!

N!

O!

LM

NO

k

mA!B!

C!

A

BC

E!

D!F!

F

ED

x

y

OS!

T!

W!

S

T

W

x

y

O

G H

K J R

G!

H!

J!

K!Q

P S

R

Q!

S!

P!


Rotate each figure about point R under the given angle of rotation and the givendirection. Label the vertices of the rotation image.

1. 80° counterclockwise 2. 100° clockwise

COORDINATE GEOMETRY Draw the rotation image of each figure 90° in the givendirection about the center point and label the coordinates.

3. !RST with vertices R("3, 3), S(2, 4), 4. !HJK with vertices H(3, 1), J(3, "3),and T(1, 2) clockwise about the and K("3, "4) counterclockwise about point P(1, 0) the point P("1, "1)

Use a composition of reflections to find the rotation image with respect to lines pand s. Then find the angle of rotation for each image.

5. 6.

7. STEAMBOATS A paddle wheel on a steamboat is driven by a steam engine and movesfrom one paddle to the next to propel the boat through the water. If a paddle wheelconsists of 18 evenly spaced paddles, identify the order and magnitude of its rotationalsymmetry.

p s

P!

R!

S!

T!

P

R

ST

ps

E!

F!

G!

E

FG

K!

J!H!

J

H

K

x

y

OP(–1, –1)

S!

R!

T!T

S

R

x

y

O P(1, 0)

QU

T S

P

R

Q!

S!

T! U!

P!

N

M P

R

M!

N!

P!

Practice Rotations

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Reading to Learn MathematicsRotations

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9-3

Pre-Activity How do some amusement rides illustrate rotations?


What are two ways that each car rotates?

Reading the Lesson

1. List all of the following types of transformations that satisfy each description: reflection,translation, rotation.a. The transformation is an isometry.b. The transformation preserves the orientation of a figure.c. The transformation is the composite of successive reflections over two intersecting

lines.d. The transformation is the composite of successive reflections over two parallel

lines.e. A specific transformation is defined by a fixed point and a specified angle.f. A specific transformation is defined by a fixed point, a fixed line, or a fixed plane.

g. A specific transformation is defined by (x, y) → (x $ a, x $ b), for fixed values of a and b.

h. The transformation is also called a slide.i. The transformation is also called a flip.j. The transformation is also called a turn.

2. Determine the order and magnitude of the rotational symmetry for each figure.

a. b.

c. d.

Helping You Remember

3. What is an easy way to remember the order and magnitude of the rotational symmetryof a regular polygon?


Finding the Center of RotationSuppose you are told that ! X!Y!Z! is the rotation image of ! XYZ, but you are not toldwhere the center of rotation is nor the measure of the angle of rotation. Can you find them? Yes,you can. Connect two pairs of correspondingvertices with segments. In the figure, the segments YY! and ZZ! are used. Draw theperpendicular bisectors, " and m, of thesesegments. The point C where " and m intersect is the center of rotation.

1. How can you find the measure of the angle of rotation in the figure above?

Locate the center of rotation for the rotation that maps WXYZ onto W!X!Y!Z!.Then find the measure of the angle of rotation.

2. 3.

Z

Z!

W

W!

X!

X

Y

Y!

center ofrotation

angle ofrotation:about 110

Z!

Z

W

W!X!

X

Y!

Y

center ofrotation

angle ofrotation:about 100

Z!

Z

C

X!X

Y!

Y

m"

Enrichment

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Study Guide and InterventionTessellations

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Regular Tessellations A pattern that covers a plane with repeating copies of one ormore figures so that there are no overlapping or empty spaces is a tessellation. A regulartessellation uses only one type of regular polygon. In a tessellation, the sum of the measuresof the angles of the polygons surrounding a vertex is 360. If a regular polygon has an interiorangle that is a factor of 360, then the polygon will tessellate.

regular tessellation tessellation Copies of a regular hexagon Copies of a regular pentagon can form a tessellation. cannot form a tessellation.

Determine whether a regular 16-gon tessellates the plane. Explain.If m"1 is the measure of one interior angle of a regular polygon, then a formula for m"1

is m"1 # '180(n

n" 2)'. Use the formula with n # 16.

m"1 # '180(n

n" 2)'

# '180(1

166

" 2)'

# 157.5

The value 157.5 is not a factor of 360, so the 16-gon will not tessellate.

Determine whether each polygon tessellates the plane. If so, draw a sample figure.

1. scalene right triangle 2. isosceles trapezoid

Determine whether each regular polygon tessellates the plane. Explain.

3. square 4. 20-gon

5. septagon 6. 15-gon

7. octagon 8. pentagon

ExercisesExercises

ExampleExample


Tessellations with Specific Attributes A tessellation pattern can contain any typeof polygon. If the arrangement of shapes and angles at each vertex in the tessellation is thesame, the tessellation is uniform. A semi-regular tessellation is a uniform tessellationthat contains two or more regular polygons.

Determine whether a kite will tessellate the plane. If so, describe the tessellation as uniform, regular, semi-regular, or not uniform.A kite will tessellate the plane. At each vertex the sum of the measures is a $ b $ b $ c, which is 360. The tessellation is uniform.

Determine whether a semi-regular tessellation can be created from each set offigures. If so, sketch the tessellation. Assume that each figure has a side length of1 unit.

1. rhombus, equilateral triangle, 2. square and equilateral triangleand octagon

Determine whether each polygon tessellates the plane. If so, describe thetessellation as uniform, not uniform, regular, or semi-regular.

3. rectangle 4. hexagon and square

c& a&

b&

b&

c&

c&

a&

a&b&

b&

b&

c&a&

b&

b&

c&a&

b&

b&

c&a&

b&

b&

b&

c& a&

b&

b&


Tessellations

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ExampleExample

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1. 15-gon 2. 18-gon

3. square 4. 20-gon

Determine whether a semi-regular tessellation can be created from each set offigures. Assume each figure has a side length of 1 unit.

5. regular pentagons and equilateral triangles

6. regular dodecagons and equilateral triangles

7. regular octagons and equilateral triangles


8. rhombus 9. isosceles trapezoid and square

Determine whether each pattern is a tessellation. If so, describe it as uniform, notuniform, regular, or semi-regular.

10. 11.



1. 22-gon 2. 40-gon


3. regular pentagons and regular decagons

4. regular dodecagons, regular hexagons, and squares


5. kite 6. octagon and decagon


7. 8.

FLOOR TILES For Exercises 9 and 10, use the following information.Mr. Martinez chose the pattern of tile shown to retile his kitchen floor.

9. Determine whether the pattern is a tessellation. Explain

10. Is the pattern uniform, regular, or semi-regular?

Practice Tessellations

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Pre-Activity How are tessellations used in art?


• In the pattern shown in the picture in your textbook, how many smallequilateral triangles make up one regular hexagon?

• In this pattern, how many fish make up one equilateral triangle?

Reading the Lesson

1. Underline the correct word, phrase, or number to form a true statement.a. A tessellation is a pattern that covers a plane with the same figure or set of figures so

that there are no (congruent angles/overlapping or empty spaces/right angles).b. A tessellation that uses only one type of regular polygon is called a

(uniform/regular/semi-regular) tessellation.c. The sum of the measures of the angles at any vertex in any tessellation is

(90/180/360).d. A tessellation that contains the same arrangement of shapes and angles at every

vertex is called a (uniform/regular/semi-regular) tessellation.e. In a regular tessellation made up of hexagons, there are (3/4/6) hexagons meeting at

each vertex, and the measure of each of the angles at any vertex is (60/90/120).f. A uniform tessellation formed using two or more regular polygons is called a

(rotational/regular/semi-regular) tessellation.g. In a regular tessellation made up of triangles, there are (3/4/6) triangles meeting at

each vertex, and the measure of each of the angles at any vertex is (30/60/120).h. If a regular tessellation is made up of quadrilaterals, all of the quadrilaterals must be

congruent (rectangles/parallelograms/squares/trapezoids).

2. Write all of the following words that describe each tessellation: uniform, non-uniform,regular, semi-regular.

a. b.

Helping You Remember

3. Often the everyday meanings of a word can help you to remember its mathematicalmeaning. Look up uniform in your dictionary. How can its everyday meanings help youto remember the meaning of a uniform tessellation?


Polygonal NumbersCertain numbers related to regular polygons are called polygonal numbers. The chartshows several triangular, square, and pentagonal numbers. The rank of a polygon numberis the number of dots on each “side” of the outer polygon. For example, the pentagonalnumber 22 has a rank of 4.

Polygonal numbers can be described with formulas. For example, a triangular number T

of rank r can be described by T # 'r(r

2$ 1)'.

Answer each question.

1. Draw a diagram to find the 2. Draw a diagram to find the triangular number of rank 5. pentagonal number of rank 5.

3. Write a formula for a square number 4. Write a formula for a pentagonalS of rank r. number P of rank r.

5. What is the rank of the pentagonal 6. List the hexagonal numbers for ranksnumber 70? 1 to 5. (Hint: Draw a diagram.)

Rank 1

Triangle

1 3 6 10

1 4 9 16

1 5 12 22

Square

Pentagon

Rank 2 Rank 3 Rank 4

Enrichment

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Study Guide and InterventionDilations

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Classify Dilations A dilation is a transformation in which the image may be adifferent size than the preimage. A dilation requires a center point and a scale factor, r.

Let r represent the scale factor of a dilation.

If | r | ( 1, then the dilation is an enlargement.

If | r | # 1, then the dilation is a congruence transformation.

If 0 ) | r | ) 1, then the dilation is a reduction.

Draw the dilation image of !ABC with center O and r $ 2.Draw OA!!", OB!!", and OC!!". Label points A!, B!, and C!so that OA! # 2(OA), OB! # 2(OB), and OC! # 2(OC). !A!B!C! is a dilation of !ABC.

Draw the dilation image of each figure with center C and the given scale factor.Describe each transformation as an enlargement, congruence, or reduction.

1. r # 2 2. r # '12'

3. r # 1 4. r # 3

5. r # '23' 6. r # 1

K J

HF

C

G

M

C

P

R

V

W

M!

P!R!

V!

W!

CS

T S!

T!

C

W

Y

XZ

C

S T

R S! T!

R!C

B

A

A!

B!

O

B

A

CO

B

A

C

A!

B!

C!

ExercisesExercises

ExampleExample


Identify the Scale Factor If you know corresponding measurements for a preimageand its dilation image, you can find the scale factor.

Determine the scale factor for the dilation of X!Y! to A!B!. Determine whether the dilation is an enlargement,reduction, or congruence transformation.

scale factor # 'priemimag

aege

lelnegntghth'

# '84

uu

nn

iittss'

# 2The scale factor is greater than 1, so the dilation is an enlargement.

Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation.

1. CGHJ is a dilation image of CDEF. 2. !CKL is a dilation image of !CKL.

3. STUVWX is a dilation image 4. !CPQ is a dilation image of !CYZ.of MNOPQR.

5. !EFG is a dilation image of !ABC. 6. !HJK is a dilation image of !HJK.

C

J

K

H

CB

DG

F

EA

C

Z

Y

P

QC

M N

O

PQ

RUX

S T

VW

C

K L

G

C

D

H

F

E

J

A

B

X

Y

C


Dilations

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ExampleExample

ExercisesExercises

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Draw the dilation image of each figure with center C and the given scale factor.

1. r # 2 2. r # '14'

Find the measure of the dilation image M!!!N!!! or of the preimage M!N! using thegiven scale factor.

3. MN # 3, r # 3 4. M!N! # 7, r # 21

COORDINATE GEOMETRY Find the image of each polygon, given the vertices, aftera dilation centered at the origin with a scale factor of 2. Then graph a dilation centered at the origin with a scale factor of '

12'.

5. J(2, 4), K(4, 4), P(3, 2) 6. D("2, 0), G(0, 2), F(2, "2)

Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation. The dashedfigure is the dilation image.

7. 8.

CC

G!

F!

D!

G%

F %

D %x

y

O

J! K!

P!J % K %

P %x

y

O

CC


Draw the dilation image of each figure with center C and the given scale factor.

1. r # '32' 2. r # '

23'

Find the measure of the dilation image A!!!T!!! or of the preimage A!T! using the givenscale factor.

3. AT # 15, r # '35' 4. AT # 30, r # "'

16' 5. A!T! # 12, r # '

43'

COORDINATE GEOMETRY Find the image of each polygon, given the vertices, aftera dilation centered at the origin with a scale factor of 2. Then graph a dilation centered at the origin with a scale factor of '

12'.

6. A(1, 1), C(2, 3), D(4, 2), E(3, 1) 7. Q("1, "1), R(0, 2), S(2, 1)

Determine the scale factor for each dilation with center C. Determine whether thedilation is an enlargement, reduction, or congruence transformation. The dottedfigure is the dilation image.

8. 9.

10. PHOTOGRAPHY Estebe enlarged a 4-inch by 6-inch photograph by a factor of '52'. What

are the new dimensions of the photograph?

CC

S!

Q!

R!

S %

Q %

R %

x

y

O

C!

D!

E!A!

C %D %

E %A%x

y

O

C

C

Practice Dilations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5

Reading to Learn MathematicsDilations

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on

9-5

Pre-Activity How do you use dilations when you use a computer?


In addition to the example given in your textbook, give two everydayexamples of scaling an object, one that makes the object larger and anotherthat makes it smaller.

Reading the Lesson1. Each of the values of r given below represents the scale factor for a dilation. In each

case, determine whether the dilation is an enlargement, a reduction, or a congruencetransformation.a. r # 3 b. r # 0.5c. r # "0.75 d. r # "1

e. r # '23' f. r # "'

32'

g. r # "1.01 h. r # 0.999

2. Determine whether each sentence is always, sometimes, or never true. If the sentence isnot always true, explain why.a. A dilation requires a center point and a scale factor.

b. A dilation changes the size of a figure.

c. A dilation changes the shape of a figure.

d. The image of a figure under a dilation lies on the opposite side of the center from thepreimage.

e. A similarity transformation is a congruence transformation.

f. The center of a dilation is its own image.

g. A dilation is an isometry.

h. The scale factor for a dilation is a positive number.

i. Dilations produce similar figures.

Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose that your

classmate Lydia is having trouble understanding the relationship between similaritytransformations and congruence transformations. How can you explain this to her?


Similar CirclesYou may be surprised to learn that two noncongruent circles that lie inthe same plane and have no common interior points can be mappedone onto the other by more than one dilation.

1. Here is diagram that suggests one way to map a smaller circle onto a larger one using a dilation. The circles are given. The linessuggest how to find the center for the dilation. Describe how the center is found.Use segments in the diagram to name the scale factor.

2. Here is another pair of noncongruent circles with no commoninterior point. From Exercise 1, you know you can locate a point off to the left of the smaller circle that is the center for a dilationmapping #C onto #C!. Find another center for another dilationthat maps #C onto #C!. Mark and label segments to name thescale factor.

X

X!

C!CO

A !

M!

M

AO

Enrichment

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Study Guide and InterventionVectors

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Magnitude and Direction A vector is a directed segment representing a quantity that has both magnitude, or length,and direction. For example, the speed and direction of an airplane can be represented by a vector. In symbols, a vector is written as AB!, where A is the initial point and B is the endpoint,or as v!.

A vector in standard position has its initial point at (0, 0) and can be represented by the ordered pair for point B. The vector at the right can be expressed as v! # "5, 3#.

You can use the Distance Formula to find the magnitude | AB! | of a vector. You can describe the direction of a vector bymeasuring the angle that the vector forms with the positivex-axis or with any other horizontal line.

Find the magnitude and direction of AB! for A(5, 2) and B(8, 7).Find the magnitude.|AB | # $(x2 "!x1)2 $! ( y2 "! y1)2!

# $(8 " 5!)2 $ (!7 " 2!)2!# $34! or about 5.8 units

To find the direction, use the tangent ratio.

tan A # '53' The tangent ratio is opposite over adjacent.

m"A% 59.0 Use a calculator.

The magnitude of the vector is about 5.8 units and its direction is 59°.

Find the magnitude and direction of AB! for the given coordinates. Round to thenearest tenth.

1. A(3, 1), B("2, 3) 2. A(0, 0), B("2, 1)

3. A(0, 1), B(3, 5) 4. A("2, 2), B(3, 1)

5. A(3, 4), B(0, 0) 6. A(4, 2), B(0, 3)

x

y

O

B(8,7)

A(5, 2)

x

y

O

B(5, 3)

A(0, 0)

V!

ExercisesExercises

ExampleExample


Translations with Vectors Recall that the transformation (a, b) → (a $ 2, b " 3)represents a translation right 2 units and down 3 units. The vector "2, "3# is another way todescribe that translation. Also, two vectors can be added: "a, b# $ "c, d# # "a $ c, b $ d#. Thesum of two vectors is called the resultant.

Graph the image of parallelogram RSTUunder the translation by the vectors m! $ "3, "1# and n! $ ""2, "4#.Find the sum of the vectors.m!$ n! # "3, "1# $ ""2, "4#

# "3 " 2, "1 " 4## "1, "5#

Translate each vertex of parallelogram RSTU right 1 unit and down 5 units.

Graph the image of each figure under a translation by the given vector(s).

1. !ABC with vertices A("1, 2), B(0, 0), 2. ABCD with vertices A("4, 1), B("2, 3),and C(2, 3); m!# "2, "3# C(1, 1), and D("1, "1); n! # "3 "3#

3. ABCD with vertices A("3, 3), B(1, 3), C(1, 1), and D("3, 1); the sum of p! # ""2, 1# and q! # "5, "4#

Given m! $ "1, "2# and n! $ ""3, "4#, represent each of the following as a singlevector.

4. m!$ n! 5. n! " m!

x

y

D!

A!B!

C!

D

A B

C

x

y

D!

A!

B!

C!D

A

B

C

x

y

A!

B!

C!

A

B

C

x

y

R!S!

T!

U!

R

S

TU


Vectors

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ExercisesExercises

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Write the component form of each vector.

1. 2.

Find the magnitude and direction of RS! for the given coordinates. Round to thenearest tenth.

3. R(2, "3), S(4, 9) 4. R(0, 2), S(3, 12)

5. R(5, 4), S("3, 1) 6. R(1, 5), S("4, "6)


7. !ABC with vertices A("4, 3), B("1, 4), 8. trapezoid with vertices T("4, "2),C("1, 1); t! # "4, "3# R("1, "2), S("2, "3), Y("3, "3); a! # "3, 1#

and b! # "2, 4#

Find the magnitude and direction of each resultant for the given vectors.

9. y! # "7, 0#, z! # "0, 6# 10. b! # "3, 2#, c! # ""2, 3#

S!Y!

T! R!

SY

T Rx

y

O

C!

B!A!C

B

A

x

y

O

x

y

O

B(–2, 2)

D(3, –3)

x

y

OC(1, –1)

E(4, 3)


Write the component form of each vector.

1. 2.

Find the magnitude and direction of FG! for the given coordinates. Round to thenearest tenth.

3. F("8, "5), G("2, 7) 4. F("4, 1), G(5, "6)


5. !QRT with vertices Q("1, 1), R(1, 4), 6. trapezoid with vertices J("4, "1),T(5, 1); s! # ""2, "5# K(0, "1), L("1, "3), M("2, "3); c! # "5, 4#

and d! # ""2, 1#

Find the magnitude and direction of each resultant for the given vectors.

7. a! # ""6, 4#, b! # "4, 6# 8. e! # ""4, "5#, f! # ""1, 3#

AVIATION For Exercises 9 and 10, use the following information.A jet begins a flight along a path due north at 300 miles per hour. A wind is blowing due westat 30 miles per hour.

9. Find the resultant velocity of the plane.

10. Find the resultant direction of the plane.

L!M!

J! K!

LM

JK

x

y

O

T!

R!

Q!

T

R

Qx

y

O

x

y

O

K(–2, 4)

L(3, –4)

x

y

O

A(–2, –2)

B(4, 2)

Practice Vectors

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6

Reading to Learn MathematicsVectors

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9-6

Pre-Activity How do vectors help a pilot plan a flight?


Why do pilots often head their planes in a slightly different direction fromtheir destination?

Reading the Lesson1. Supply the missing words or phrases to complete the following sentences.

a. A is a directed segment representing a quantity that has bothmagnitude and direction.

b. The length of a vector is called its .c. Two vectors are parallel if and only if they have the same or

direction.d. A vector is in if it is drawn with initial point at the origin.e. Two vectors are equal if and only if they have the same and the

same .f. The sum of two vectors is called the .g. A vector is written in if it is expressed as an ordered pair.h. The process of multiplying a vector by a constant is called .

2. Write each vector described below in component form.

a. a vector in standard position with endpoint (a, b)

b. a vector with initial point (a, b) and endpoint (c, d)

c. a vector in standard position with endpoint ("3, 5)

d. a vector with initial point (2, "3) and endpoint (6, "8)

e. a! $ b! if a! # ""3, 5# and b! # "6, "4#f. 5u! if u! # "8, "6#g. "'

13'v! if v! # ""15, 24#

h. 0.5u! $ 1.5v! if u! # "10, "10# and v! # ""8, 6#

Helping You Remember3. A good way to remember a new mathematical term is to relate it to a term you already

know. You learned about scale factors when you studied similarity and dilations. How is theidea of a scalar related to scale factors?


Reading MathematicsMany quantities in nature can be thought of as vectors. The science ofphysics involves many vector quantities. In reading about applicationsof mathematics, ask yourself whether the quantities involve onlymagnitude or both magnitude and direction. The first kind of quantityis called scalar. The second kind is a vector.

Classify each of the following. Write scalar or vector.

1. the mass of a book

2. a car traveling north at 55 mph

3. a balloon rising 24 feet per minute

4. the size of a shoe

5. a room temperature of 22 degrees Celsius

6. a west wind of 15 mph

7. the batting average of a baseball player

8. a car traveling 60 mph

9. a rock falling at 10 mph

10. your age

11. the force of Earth’s gravity acting on a moving satellite

12. the area of a record rotating on a turntable

13. the length of a vector in the coordinate plane

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6

Study Guide and InterventionTransformations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

9-79-7


Less

on

9-7

Translations and Dilations A vector can be represented by the ordered pair "x, y# or

by the column matrix . When the ordered pairs for all the vertices of a polygon are

placed together, the resulting matrix is called the vertex matrix for the polygon.

For !ABC with A("2, 2), B(2, 1), and C("1, "1), the vertex

matrix for the triangle is .

For !ABC above, use a matrix to find the coordinates of thevertices of the image of !ABC under the translation (x, y) → (x # 3, y " 1).To translate the figure 3 units to the right, add 3 to each x-coordinate. To translate thefigure 1 unit down, add "1 to each y-coordinate.

Vertex Matrix Translation Vertex Matrixof !ABC Matrix of !A!B!C!

$ #

The coordinates are A!(1, 1), B!(5, 0), and C!(2, "2).

For !ABC above, use a matrix to find the coordinates of the verticesof the image of !ABC for a dilation centered at the origin with scale factor 3.

Scale Vertex Matrix Vertex MatrixFactor of !ABC of !A!B!C!

3 * #

The coordinates are A!("6, 6), B!(6, 3), and C!("3, "3).

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations or dilations.

1. !ABC with A(3, 1), B("2, 4), C("2, "1); (x, y) → (x " 1, y $ 2)

2. parallelogram RSTU with R("4, "2), S("3, 1), T(3, 4), U(2, 1); (x, y) → (x " 4, y " 3)

3. rectangle PQRS with P(4, 0), Q(3, "3), R("3, "1), S("2, 2); (x, y) → (x " 2, y $ 1)

4. !ABC with A("2, "1), B("2, "3), C(2, "1); dilation centered at the origin with scalefactor 2

5. parallelogram RSTU with R(4, "2), S("4, "1), T("2, 3), U(6, 2); dilation centered at theorigin with scale factor 1.5

⎡"6 6 "3⎤⎣ 6 3 "3⎦

⎡"2 2 "1⎤⎣ 2 1 "1⎦

⎡1 5 2⎤⎣1 0 "2⎦

⎡ 3 3 3⎤⎣"1 "1 "1⎦

⎡"2 2 "1⎤⎣ 2 1 "1⎦

⎡"2 2 "1⎤⎣ 2 1 "1⎦

x

yB(2, 1)

C(–1, –1)

A(–2, 2)

⎡ x⎤⎣ y⎦

ExercisesExercises

Example 1Example 1

Example 2Example 2


Reflections and Rotations When you reflect an image, one way to find thecoordinates of the reflected vertices is to multiply the vertex matrix of the object by areflection matrix. To perform more than one reflection, multiply by one reflection matrixto find the first image. Then multiply by the second matrix to find the final image. Thematrices for reflections in the axes, the origin, and the line y # x are shown below.

For a reflection in the: x-axis y-axis origin line y $ x

Multiply the vertex matrix by:

!ABC has vertices A("2, 3), B(1, 4), and C(3, 0). Use a matrix to find the coordinates of the vertices of the image of !ABC after a reflection in the x-axis.To reflect in the x-axis, multiply the vertex matrix of !ABC by thereflection matrix for the x-axis.

Reflection Matrix Vertex Matrix Vertex Matrix for x-axis of !ABC of !A’B’C’

* #

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given reflection.

1. !ABC with A("3, 2), B("1, 3), C(1, 0); reflection in the x-axis

2. !XYZ with X(2, "1), Y(4, "3), Z("2, 1); reflection in the y-axis

3. !ABC with A(3, 4), B("1, 0), C("2, 4); reflection in the origin

4. parallelogram RSTU with R("3, 2), S(3, 2), T(5, "1), U("1, "1); reflection in the line y # x

5. !ABC with A(2, 3), B("1, 2), C(1, "1); reflection in the origin, then reflection in the liney # x

6. parallelogram RSTU with R(0, 2), S(4, 2), T(3, "2), U("1, "2); reflection in the x-axis,then reflection in the y-axis

⎡"2 1 3⎤⎣"3 "4 0⎦

⎡"2 1 3⎤⎣ 3 4 0⎦

⎡1 0⎤⎣0 "1⎦

x

y

A!B!

AB

C

⎡0 1⎤⎣1 0⎦

⎡"1 0⎤⎣ 0 "1⎦

⎡"1 0⎤⎣ 0 1⎦

⎡1 0⎤⎣0 "1⎦


Transformations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

9-79-7

ExampleExample

ExercisesExercises

Skills PracticeTransformations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

9-79-7


Less

on

9-7

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations.

1. !STU with S(6, 4), T(9, 7), and U(14, 2); (x, y) → (x " 4, y $ 3)

2. !GHI with G("5, 0), H("3, 6), and I("2, 1); (x, y) → (x $ 2, y $ 6)

Use scalar multiplication to find the coordinates of the vertices of each figure fora dilation centered at the origin with the given scale factor.

3. !DEF with D(2, 1), E(5, 4), and F(7, 2); r # 4

4. quadrilateral WXYZ with W("9, 6), X("6, 3), Y(3, 12), and Z("6, 15); r # '13'


5. !MNO with M("5, 1), N("2, 3), and O(2, 0); y-axis

6. quadrilateral ABCD with A(3, 1), B(6, "2), C(5, "5), and D(1, "6); x-axis

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given rotation.

7. !RST with R("2, "2), S("3, 3), and T(2, 2); 90°counterclockwise

8. $LMNP with L(3, 4), M(7, 4), N(9, "3), and P(5, "3); 180° counterclockwise


Use a matrix to find the coordinates of the vertices of the image of each figureunder the given translations.

1. !KLM with K("7, "3), L(4, 9), and M(9, "6); (x, y) → (x " 7, y $ 2)

2. $ABCD with A("4, 3), B("2, 8), C(3, 10), and D(1, 5); (x, y) → (x $ 3, y " 9)

Use scalar multiplication to find the coordinates of the vertices of each figure fora dilation centered at the origin with the given scale factor.

3. quadrilateral HIJK with H("2, 3), I(2, 6), J(8, 3), and K(3, "4); r # "'13'

4. pentagon DEFGH with D("8, "4), E("8, 2), F(2, 6), G(8, 0), and H(4, "6); r # '54'


5. !QRS with Q("5, "4), R("1, "1), and S(2, "6); x-axis

6. quadrilateral VXYZ with V("4, "2), X("3, 4), Y(2, 1), and Z(4, "3); y # x

Use a matrix to find the coordinates of the vertices of the image of each figureunder the given rotation.

7. $EFGH with E("5 "4), F("3, "1), G(5, "1), and H(3, "4); 90° counterclockwise

8. quadrilateral PSTU with P("3, 5), S(2, 6), T(8, 1), and U("6, "4); 270° counterclockwise

9. FORESTRY A research botanist mapped a section of forested land on a coordinate gridto keep track of endangered plants in the region. The vertices of the map are A("2, 6),B(9, 8), C(14, 4), and D(1, "1). After a month, the botanist has decided to decrease the research area to '

34' of its original size. If the center for the reduction is O(0, 0),what are

the coordinates of the new research area?

Practice Transformations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

9-79-7

Reading to Learn MathematicsTransformations with Matrices

NAME ______________________________________________ DATE ____________ PERIOD _____

9-79-7


Less

on

9-7

Pre-Activity How can matrices be used to make movies?


• What kind of transformation should be used to move a polygon?

• What kind of transformation should be used to resize a polygon?

Reading the Lesson1. Write a vertex matrix for each figure.

a. !ABC b. parallelogram ABCD

2. Match each transformation from the first column with the corresponding matrix fromthe second or third column. In each case, the vertex matrix for the preimage of a figureis multiplied on the left by one of the matrices below to obtain the image of the figure.All rotations listed are counterclockwise through the origin. (Some matrices may be usedmore than once or not at all.)

a. reflection over the y-axis i. v.b. 90° rotationc. reflection over the line y # x ii. vi.d. 270° rotatione. reflection over the origin iii. vii.f. 180° rotationg. reflection over the x-axis iv. viii.h. 360° rotation

Helping You Remember3. How can you remember or quickly figure out the matrices for the transformations in

Exercise 2?

⎡0 "1⎤⎣1 0⎦

⎡ 0 "1⎤⎣"1 0⎦

⎡1 0⎤⎣0 "1⎦

⎡0 1⎤⎣1 0⎦

⎡"1 0⎤⎣ 0 1⎦

⎡"1 0⎤⎣ 0 "1⎦

⎡ 0 1⎤⎣"1 0⎦

⎡1 0⎤⎣0 1⎦

B C

A D x

y

O

B

C

A

x

y

O


Vector AdditionVectors are physical quantities with magnitude and direction. Force and velocity are twoexamples. We will investigate adding vector quantities. The sum of two vectors is called aresultant vector or just the resultant.

Two separate forces, one measuring 20 units and the other measuring 40 units, act on an object. If the angle between the forces is 50°, find themagnitude and direction of the resultant force.

First, the vectors must be rearranged by placing the tail of the 20-unit vector at the head of the 40-unit vector. Since these vectors are not perpendicular, the horizontal and verticalcomponents of one of the vectors must be found. Using trigonometry, the horizontal component must be (20 cos 50°) units and the vertical component must be (20 sin 50°) units.Replacing the 20-unit vector with these components, we can now form two vectors perpendicular and use the Pythagorean Theorem to find the resultant.

r2 # (40 $ 20 cos 50°)2 $ (20 sin 50°)2

r2 % (52.9)2 $ (15.3)2

r2 % 3032.5r2 % 55.1

tan O # '402$0

2s0in

c5os0°

50°'

% 0.2898m"O % 16

Therefore, the resultant force is 55.1 units directed 16° from the 40-unit force.

Solve. Round all angle measures to the nearest degree. Round all other measuresto the nearest tenth.

1. A plane flies due west at 250 kilometers per hour while the wind blows south at 70kilometers per hour. Find the plane’s resultant velocity.

2. A plane flies east for 200 km, then 60° south of east for 80 km. Find the plane’s distanceand direction from its starting point.

3. One force of 100 units acts on an object. Another force of 80 units acts on the object at a40° angle from the first force. Find the resultant force on the object.

O

20

40 # 20 cos 50&

20 sin 50&r

20

20 cos 50&

20 sin 50&

50&

O

20 units

40 units50&

O

20 units

40 units50&

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-79-7

ExampleExample

© Glencoe/McGraw-Hill A2 Glencoe Geometry

Stu

dy

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Ref

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9-1

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Lesson 9-1

Dra

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calle

d a

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onis

a f

lip o

f a

figu

re in

a

poin

t,a

line,

or a

pla

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he n

ew f

igur

e is

the

im

age

and

the

orig

inal

fig

ure

is t

hep

reim

age.

The

pre

imag

e an

d im

age

are

cong

ruen

t,so

a r

efle

ctio

n is

a c

ongr

uen

cetr

ansf

orm

atio

nor

iso

met

ry.

Con

stru

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he

imag

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qu

adri

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AB

CD

un

der

are

flec

tion

in

lin

em

.

Dra

w a

per

pend

icul

ar fr

om e

ach

vert

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B!,

C!,

and

D!

that

are

the

sam

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A!B

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m

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B

A!

B!

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D!

Qu

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DE

FG

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F(3

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,an

d

G("

3,"

1).F

ind

th

e im

age

un

der

ref

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in t

he

x-ax

is.

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n im

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for

are

flec

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in t

he x

-axi

s,us

e th

e sa

me

x-co

ordi

nate

and

mul

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e y-

coor

dina

te b

y "

1.In

sym

bols

,(a,

b) →

(a,"

b).

The

new

coo

rdin

ates

are

D!(

"2,

"3)

,E!(

4,"

4),

F!(

3,2)

,and

G!(

"3,

1).

The

imag

e is

D!E

!F!G

!.

x

y

O

D!

E!

F!

G!D

E

FG

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

In E

xam

ple

2,th

e no

tati

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epre

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the

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in t

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1.2.

3.

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4.!

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5.A

BC

Dw

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A(1

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B(3

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C(2

,"2)

,F

(3,"

1) in

the

x-a

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D("

3,1)

in t

he y

-axi

s

x

y

OD

!

C!

B!

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D

C

B

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Po

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wh

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here

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Det

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4.5.

6.

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9.

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9-1

9-1

Exam

ple

Exam

ple

Exer

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Exer

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Answers (Lesson 9-1)


An

swer

s

Skil

ls P

ract

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ions

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9-1

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Lesson 9-1

Dra

w t

he

imag

e of

eac

h f

igu

re u

nd

er a

ref

lect

ion

in

lin

e !.

1.2.

CO

OR

DIN

ATE

GEO

MET

RYG

rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

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3.!

AB

Cw

ith

vert

ices

A("

3,2)

,B(0

,1),

4.tr

apez

oid

DE

FG

wit

h ve

rtic

es D

(0,"

3),

and

C("

2,"

3) in

the

ori

gin

E(1

,3),

F(3

,3),

and

G(4

,"3)

in t

he y

-axi

s

5.pa

ralle

logr

am R

ST

Uw

ith

vert

ices

6.

squa

re K

LM

Nw

ith

vert

ices

K("

1,0)

,R

("2,

3),S

(2,4

),T

(2,"

3) a

nd

L("

2,3)

,M(1

,4),

and

N(2

,1)

in

U("

2,"

4) in

the

line

y#

xth

e x-

axis

Det

erm

ine

how

man

y li

nes

of

sym

met

ry e

ach

fig

ure

has

.Th

en d

eter

min

e w

het

her

the

figu

re h

as p

oin

t sy

mm

etry

.

7.8.

9.

1;no

4;ye

s0;

yes

M!

K!

Kx

y

O

L!

N!

LM

Nx

y

O

R!S

!T

!

U!

RS T

U

x

y

O

DEF

GD

!

E!

F!

G!

x

y

O

A!

B!

C!

AB

C

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er a

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1.2.

CO

OR

DIN

ATE

GEO

MET

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rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

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.

3.qu

adri

late

ral A

BC

Dw

ith

vert

ices

4.

!F

GH

wit

h ve

rtic

es F

("3,

"1)

,G(0

,4),

A("

3,3)

,B(1

,4),

C(4

,0),

and

and

H(3

,"1)

in t

he li

ne y

#x

D("

3,"

3) in

the

ori

gin

5.re

ctan

gle

QR

ST

wit

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rtic

es Q

("3,

2),

6.tr

apez

oid

HIJ

Kw

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ices

H("

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("1,

4),S

(2,1

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d T

(0,"

1)

I(2,

5),J

("4,

"1)

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K("

4,3)

in

the

x-a

xis

in t

he y

-axi

s

RO

AD

SIG

NS

Det

erm

ine

how

man

y li

nes

of

sym

met

ry e

ach

sig

n h

as.T

hen

det

erm

ine

wh

eth

er t

he

sign

has

poi

nt

sym

met

ry.

7.8.

9.

0;ye

s1;

no4;

yes

HI

JK

H!

I!

J!K! x

y

OQ

!

R!

S!

T!

Q

R

S

Tx

y

O

F!

G!

H!

F

G

Hx

y

O

A!

B!

C!

D!

AB

C

D

x

y

O

!!Pra

ctic

e (A

vera

ge)

Ref

lect

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-1

9-1



Rea

din

g t

o L

earn

Math

emati

csR

efle

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-1

9-1

©G

lenc

oe/M

cGra

w-H

ill48

3G

lenc

oe G

eom

etry

Lesson 9-1

Pre-

Act

ivit

yW

her

e ar

e re

flec

tion

s fo

un

d i

n n

atu

re?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 9-

1 at

the

top

of

page

463

in y

our

text

book

.

Supp

ose

you

draw

a li

ne s

egm

ent

conn

ecti

ng a

poi

nt a

t th

e pe

ak o

f a m

ount

ain

to it

s im

age

in t

he la

ke.W

here

will

the

mid

poin

t of

thi

s se

gmen

t fa

ll?on

the

boun

dary

line

bet

wee

n th

e sh

ore

and

the

surf

ace

of t

he la

ke

Rea

din

g t

he

Less

on

1.D

raw

the

ref

lect

ed im

age

for

each

ref

lect

ion

desc

ribe

d be

low

.

a.re

flec

tion

of

trap

ezoi

d A

BC

Din

the

line

nb.

refl

ecti

on o

f !R

ST

in p

oint

PL

abel

the

imag

e of

AB

CD

as A

!B!C

!D!.

Lab

el t

he im

age

of R

ST

as R

!S!T

!.

c.re

flec

tion

of

pent

agon

AB

CD

Ein

the

ori

gin

Lab

el t

he im

age

of A

BC

DE

as A

!B!C

!D!E

!.

2.D

eter

min

e th

e im

age

of t

he g

iven

poi

nt u

nder

the

indi

cate

d re

flec

tion

.a.

(4,6

);re

flec

tion

in t

he y

-axi

s("

4,6)

b.("

3,5)

;ref

lect

ion

in t

he x

-axi

s("

3,"

5)c.

("8,

"2)

;ref

lect

ion

in t

he li

ney

#x

("2,

"8)

d.(9

,"3)

;ref

lect

ion

in t

he o

rigi

n("

9,3)

3.D

eter

min

e th

e nu

mbe

r of

line

s of

sym

met

ry f

or e

ach

figu

re d

escr

ibed

bel

ow.T

hen

dete

rmin

e w

heth

er t

he f

igur

e ha

s po

int

sym

met

ry a

nd in

dica

te t

his

by w

riti

ng y

esor

no.

a.a

squa

re 4

;ye

sb.

an is

osce

les

tria

ngle

(no

t eq

uila

tera

l) 1

;no

c.a

regu

lar

hexa

gon

6;ye

sd.

an is

osce

les

trap

ezoi

d 1;

noe.

a re

ctan

gle

(not

a s

quar

e) 2

;ye

sf.

the

lett

er E

1;

no

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

geo

met

ric

term

is t

o re

late

the

wor

d or

its

part

s to

geom

etri

c te

rms

you

alre

ady

know

.Loo

k up

the

ori

gins

of

the

two

part

s of

the

wor

dis

omet

ryin

you

r di

ctio

nary

.Exp

lain

the

mea

ning

of

each

par

t an

d gi

ve a

ter

m y

oual

read

y kn

ow t

hat

shar

es t

he o

rigi

n of

tha

t pa

rt.

Sam

ple

answ

er:T

he f

irst

par

tco

mes

fro

m is

os,w

hich

mea

ns e

qual

,as

in is

osce

les.

The

seco

nd p

art

com

es f

rom

met

ron,

whi

ch m

eans

mea

sure

,as

in g

eom

etry

.

A!

B!

C!

D!

E!

AB

CD

E

x

y

O

R!

S!

T!

RS

P

Tn

A!

B!

C!

D!

A

BC D

©G

lenc

oe/M

cGra

w-H

ill48

4G

lenc

oe G

eom

etry

Ref

lect

ions

in th

e C

oord

inat

e P

lane

Stu

dy

the

dia

gram

at

the

righ

t.It

sh

ows

how

th

e tr

ian

gle

AB

Cis

map

ped

on

to t

rian

gle

XY

Zby

th

etr

ansf

orm

atio

n (

x,y)

→("

x#

6,y)

.Not

ice

that

!X

YZ

is t

he

refl

ecti

on i

mag

e w

ith

res

pec

t to

th

e ve

rtic

al

lin

e w

ith

equ

atio

n x

$3.

1.P

rove

tha

t th

e ve

rtic

al li

ne w

ith

equa

tion

x#

3 is

the

perp

endi

cula

r bi

sect

or o

f th

e se

gmen

t w

ith

endp

oint

s (x

,y)

and

("x

$6,

y).(

Hin

t:U

se t

he m

idpo

int

form

ula.

)

Mid

poin

t $

!%x#

("2x

#6)

%,%

y# 2

y%

"or

(3,y

)

The

segm

ent

join

ing

( x,y

) an

d ("

x#

6,y

) is

ho

rizo

ntal

and

hen

ce is

per

pend

icul

ar t

o th

e ve

rtic

al li

ne.

2.E

very

tra

nsfo

rmat

ion

of t

he f

orm

(x,

y) →

("x

$2

h,y)

is

a re

flec

tion

wit

h re

spec

t to

the

ver

tica

l lin

e w

ith

equa

tion

x

#h.

Wha

t ki

nd o

f tr

ansf

orm

atio

n is

(x,

y) →

(x,"

y$

2k)

?

refle

ctio

n w

ith r

espe

ct t

o th

e ho

rizo

ntal

line

with

equ

atio

n y

$k

Dra

w t

he

tran

sfor

mat

ion

im

age

for

each

fig

ure

an

d t

he

give

n

tran

sfor

mat

ion

.Is

it a

ref

lect

ion

tra

nsf

orm

atio

n?

If s

o,w

ith

re

spec

t to

wh

at l

ine?

3.(x

,y) →

("x

"4,

y)4.

(x,y

) →(x

,"y

$8)

yes;

x$

"2

yes;

y$

4

x

y

Ox

y

O

x $

3

A

B

C

X

Y

Z

(x, y

) → ("

x #

6, y

)

x

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-1

9-1



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Tran

slat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-2

9-2

©G

lenc

oe/M

cGra

w-H

ill48

5G

lenc

oe G

eom

etry

Lesson 9-2

Tran

slat

ion

s U

sin

g C

oo

rdin

ates

A t

rans

form

atio

n ca

lled

a tr

ansl

atio

nsl

ides

afi

gure

in a

giv

en d

irec

tion

.In

the

coor

dina

te p

lane

,a t

rans

lati

on m

oves

eve

ry p

reim

age

poin

t P

(x,y

) to

an

imag

e po

int

P(x

$a,

y$

b) f

or f

ixed

val

ues

aan

d b.

In w

ords

,atr

ansl

atio

n sh

ifts

a f

igur

e a

unit

s ho

rizo

ntal

ly a

nd b

unit

s ve

rtic

ally

;in

sym

bols

,(x

,y) →

(x$

a,y

$b)

.

Rec

tan

gle

RE

CT

has

ver

tice

s R

("2,

"1)

,E

("2,

2),C

(3,2

),an

d T

(3,"

1).G

rap

h R

EC

Tan

d i

ts i

mag

e fo

r th

e tr

ansl

atio

n (

x,y)

→(x

#2,

y"

1).

The

tra

nsla

tion

mov

es e

very

poi

nt o

f th

e pr

eim

age

righ

t 2

unit

s an

d do

wn

1 un

it.

(x,y

) →(x

$2,

y"

1)R

("2,

"1)

→R

!("

2 $

2,"

1 "

1) o

r R

!(0,

"2)

E("

2,2)

→E

!("

2 $

2,2

"1)

or

E!(

0,1)

C(3

,2)

→C

!(3

$2,

2 "

1) o

r C

!(5,

1)T

(3,"

1) →

T!(

3 $

2,"

1 "

1) o

r T

!(5,

"2)

Gra

ph

eac

h f

igu

re a

nd

its

im

age

un

der

th

e gi

ven

tra

nsl

atio

n.

1.P !Q!

wit

h en

dpoi

nts

P("

1,3)

and

Q(2

,2)

unde

r th

e tr

ansl

atio

n le

ft 2

uni

ts a

nd u

p 1

unit

2.!

PQ

Rw

ith

vert

ices

P("

2,"

4),Q

("1,

2),a

nd R

(2,1

) un

der

the

tran

slat

ion

righ

t 2

unit

s an

d do

wn

2 un

its

3.sq

uare

SQ

UR

wit

h ve

rtic

es S

(0,2

),Q

(3,1

),U

(2,"

2),a

nd

R("

1,"

1) u

nder

the

tra

nsla

tion

rig

ht 3

uni

ts a

nd u

p 1

unit

x

y

R!S

!Q

!

U!

R

SQ

U

x

y P!

Q!

R!

P

QR

x

yP

!Q

!P

Q

x

y

OE!

R!

C!

T!

E R

C T

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill48

6G

lenc

oe G

eom

etry

Tran

slat

ion

s b

y R

epea

ted

Ref

lect

ion

sA

noth

er w

ay t

o fi

nd t

he im

age

of a

tran

slat

ion

is t

o re

flec

t th

e fi

gure

tw

ice

in p

aral

lel l

ines

.Thi

s ki

nd o

f tr

ansl

atio

n is

cal

led

aco

mp

osit

e of

ref

lect

ion

s.

In t

he

figu

re, m

|| n.F

ind

th

e tr

ansl

atio

n i

mag

e of

!A

BC

.!

A!B

!C!

is t

he im

age

of !

AB

Cre

flec

ted

in li

ne m

.!

A%B

%C%

is t

he im

age

of !

A!B

!C!

refl

ecte

d in

line

n.

The

fin

al im

age,

!A

%B%C

%,is

a t

rans

lati

on o

f !A

BC

.

In e

ach

fig

ure

, m|| n

.Fin

d t

he

tran

slat

ion

im

age

of e

ach

fig

ure

by

refl

ecti

ng

it i

nli

ne

man

d t

hen

in

lin

e n.

1.2.

3.4.

5.6.

m

U

A D

QQ

&

U&

A& D

&

U!

A!

D!

n

m n

U

T

S

R U&

T&

S&

R&

m

n

D!

Q!

AQ

U

D

A&

Q&

D&

U&

T!

mn

P!

PP

&

E!

EE

&N

N&

TT

&

AA

&

mn

A!

AB

C

A&

B&

C&

mn

A!

B!

C!

A

B C

A&

B&

C&

mn

A!

B!

C!

A

B C

A&

B&

C&

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Tran

slat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-2

9-2

Exer

cises

Exer

cises

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Tran

slat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-2

9-2

©G

lenc

oe/M

cGra

w-H

ill48

7G

lenc

oe G

eom

etry

Lesson 9-2

In e

ach

fig

ure

,a|| b

.Det

erm

ine

wh

eth

er f

igu

re 3

is

a tr

ansl

atio

n i

mag

e of

fig

ure

1.

Wri

te y

esor

no.

Exp

lain

you

r an

swer

.

1.2.

Yes;

refle

ct f

igur

e 1

in li

ne a

toN

o;it

is o

rien

ted

diff

eren

tly t

han

get

figur

e 2

and

figur

e 2

in li

ne b

figur

e 1.

to g

et f

igur

e 3.

3.4.

Yes;

refle

ct f

igur

e 1

in li

ne a

to

No;

it is

a r

efle

ctio

n of

a t

rans

latio

n ge

t fig

ure

2 an

d fig

ure

2 in

line

ban

d is

ori

ente

d di

ffer

ently

tha

n to

get

fig

ure

3.fig

ure

1.

CO

OR

DIN

ATE

GEO

MET

RYG

rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

ntr

ansl

atio

n.

5.!

JKL

wit

h ve

rtic

es J

("4,

"4)

,6.

quad

rila

tera

l LM

NP

wit

h ve

rtic

es L

(4,2

),K

("2,

"1)

,and

L(2

,"4)

und

er t

he

M(4

,"1)

,N(0

,"1)

,and

P(1

,4)

unde

r th

e tr

ansl

atio

n (x

,y) →

(x$

2,y

$5)

tran

slat

ion

(x,y

) →(x

"4,

y"

3)

M!

L MN

P

L !

N!P

!

x

y

OJ!

K!

L!

J

K

L

x

y

O

ab

12

3

ab

1

23

a b

1 2 3a

b

12

3

©G

lenc

oe/M

cGra

w-H

ill48

8G

lenc

oe G

eom

etry

In e

ach

fig

ure

,c|| d

.Det

erm

ine

wh

eth

er f

igu

re 3

is

a tr

ansl

atio

n i

mag

e of

fig

ure

1.

Wri

te y

esor

no.

Exp

lain

you

r an

swer

.

1.2.

No;

it is

a t

rans

latio

n of

a

Yes;

it is

a r

efle

ctio

n of

a r

efle

ctio

n re

flect

ion

and

is o

rien

ted

with

res

pect

to

the

para

llel l

ines

.di

ffer

ently

tha

n fig

ure

1.

CO

OR

DIN

ATE

GEO

MET

RYG

rap

h e

ach

fig

ure

an

d i

ts i

mag

e u

nd

er t

he

give

ntr

ansl

atio

n.

3.qu

adri

late

ral T

UW

Xw

ith

vert

ices

4.

pent

agon

DE

FG

Hw

ith

vert

ices

D("

1,"

2),

T("

1,1)

,U(4

,2),

W(1

,5),

and

X("

1,3)

E

(2,"

1),F

(5,"

2),G

(4,"

4),H

(1,"

4)

unde

r th

e tr

ansl

atio

n un

der

the

tran

slat

ion

(x,y

) →(x

"2,

y"

4)(x

,y) →

(x"

1,y

$5)

AN

IMA

TIO

NF

ind

th

e tr

ansl

atio

n t

hat

mov

es t

he

figu

re

on t

he

coor

din

ate

pla

ne.

5.fi

gure

1 →

figu

re 2

(x#

4,y

#2)

6.fi

gure

2 →

figu

re 3

(x"

4,y

#1)

7.fi

gure

3 →

figu

re 4

(x#

3,y

#3)

x

y

O

13

4

2

D!

E!

F!

G!

H!

DE

F

GH

x

y

O

W

T!

U!

W!

X!

TU

X

x

y

O

c d

1 2 3

cd

12

3

Pra

ctic

e (A

vera

ge)

Tran

slat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-2

9-2



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csTr

ansl

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-2

9-2

©G

lenc

oe/M

cGra

w-H

ill48

9G

lenc

oe G

eom

etry

Lesson 9-2

Pre-

Act

ivit

yH

ow a

re t

ran

slat

ion

s u

sed

in

a m

arch

ing

ban

d s

how

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 9-

2 at

the

top

of

page

470

in y

our

text

book

.

How

do

band

dir

ecto

rs g

et t

he m

arch

ing

band

to

mai

ntai

n th

e sh

ape

of t

hefi

gure

the

y or

igin

ally

for

med

?S

ampl

e an

swer

:The

ban

d pr

actic

esm

arch

ing

in u

niso

n,w

ith e

very

one

mak

ing

iden

tical

mov

es a

tth

e sa

me

time.

Rea

din

g t

he

Less

on

1.U

nder

line

the

corr

ect

wor

d or

phr

ase

to f

orm

a t

rue

stat

emen

t.a.

All

refl

ecti

ons

and

tran

slat

ions

are

(op

posi

tes/

isom

etri

es/e

quiv

alen

t).

b.T

he p

reim

age

and

imag

e of

a f

igur

e un

der

a re

flec

tion

in a

line

hav

e (t

he s

ame

orie

ntat

ion/

oppo

site

ori

enta

tion

s).

c.T

he p

reim

age

and

imag

e of

a f

igur

e un

der

a tr

ansl

atio

n ha

ve

(the

sam

e or

ient

atio

n/op

posi

te o

rien

tati

ons)

.d.

The

res

ult

of s

ucce

ssiv

e re

flec

tion

s ov

er t

wo

para

llel l

ines

is a

(ref

lect

ion/

rota

tion

/tra

nsla

tion

).e.

Col

linea

rity

(is

/is

not)

pre

serv

ed b

y tr

ansl

atio

ns.

f.T

he t

rans

lati

on (x

,y) →

(x$

a,y

$b)

shi

fts

ever

y po

int

aun

its

(hor

izon

tally

/ver

tica

lly)

and

yun

its

(hor

izon

tally

/ver

tica

lly).

2.F

ind

the

imag

e of

eac

h pr

eim

age

unde

r th

e in

dica

ted

tran

slat

ion.

a.(x

,y);

5 un

its

righ

t an

d 3

unit

s up

(x#

5,y

#3)

b.(x

,y);

2 un

its

left

and

4 u

nits

dow

n(x

"2,

y"

4)c.

(x,y

);1

unit

left

and

6 u

nits

up

(x"

1,y

#6)

d.(x

,y);

7 un

its

righ

t(x

#7,

y)e.

(4,"

3);3

uni

ts u

p(4

,0)

f.("

5,6)

;3 u

nits

rig

ht a

nd 2

uni

ts d

own

("2,

4)g.

("7,

5);7

uni

ts r

ight

and

5 u

nits

dow

n(0

,0)

h.

("9,

"2)

;12

unit

s ri

ght

and

6 un

its

dow

n(3

,"8)

3.!

RS

Tha

s ve

rtic

es R

("3,

3),S

(0,"

2),a

nd T

(2,1

).G

raph

!R

ST

and

its

imag

e !

R!S

!T!

unde

r th

e tr

ansl

atio

n (x

,y) →

(x$

3,y

"2)

.L

ist

the

coor

dina

tes

of t

he v

erti

ces

of t

he im

age.

R!(

0,1)

,S!(

3,"

4),T

!(5,

"1)

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is t

o re

late

it t

o an

eve

ryda

y m

eani

ngof

the

sam

e w

ord.

How

is t

he m

eani

ng o

f tra

nsla

tion

in g

eom

etry

rel

ated

to

the

idea

of

tran

slat

ion

from

one

lang

uage

to

anot

her?

Sam

ple

answ

er:W

hen

you

tran

slat

efr

om o

ne la

ngua

ge t

o an

othe

r,yo

u ca

rry

over

the

mea

ning

fro

m o

nela

ngua

ge to

ano

ther

.Whe

n yo

u tr

ansl

ate

a ge

omet

ric

figur

e,yo

u ca

rry

over

the

figur

e fr

om o

ne p

ositi

on to

ano

ther

with

out c

hang

ing

its b

asic

R!

S!

T!

R

S

T

x

y

O

©G

lenc

oe/M

cGra

w-H

ill49

0G

lenc

oe G

eom

etry

Tran

slat

ions

in T

he C

oord

inat

e P

lane

You

can

use

alg

ebra

ic d

escr

ipti

ons

of r

efle

ctio

ns

to s

how

th

atth

e co

mp

osit

e of

tw

o re

flec

tion

s w

ith

res

pec

t to

par

alle

l li

nes

is

a tr

ansl

atio

n (

that

is,

a sl

ide)

.

1.Su

ppos

e a

and

bar

e tw

o di

ffer

ent

real

num

bers

.Let

San

d T

be t

hefo

llow

ing

refl

ecti

ons.

S:(

x,y)

→("

x$

2a,

y)T

:(x,

y) →

("x

$2

b,y)

Sis

ref

lect

ion

wit

h re

spec

t to

the

line

wit

h eq

uati

on x

#a,

and

Tis

refl

ecti

on w

ith

resp

ect

to t

he li

ne w

ith

equa

tion

x#

b.

a.F

ind

an a

lgeb

raic

des

crip

tion

(si

mila

r to

tho

se a

bove

for

San

dT

) to

des

crib

e th

e co

mpo

site

tra

nsfo

rmat

ion

“Sfo

llow

ed b

y T

.”

(x,y

) →

(x#

2 (b

"a)

,y)

b.F

ind

an a

lgeb

raic

des

crip

tion

for

the

com

posi

te t

rans

form

atio

n“T

follo

wed

by

S.”

(x,y

) →

(x#

2 (a

"b)

,y)

2.T

hink

abo

ut t

he r

esul

ts y

ou o

btai

ned

in E

xerc

ise

1.W

hat

do t

hey

tell

you

abou

t ho

w t

he d

ista

nce

betw

een

two

para

llel l

ines

isre

late

d to

the

dis

tanc

e be

twee

n a

prei

mag

e an

d im

age

poin

t fo

r a

com

posi

te o

f re

flec

tion

s w

ith

resp

ect

to t

hese

line

s?

The

dist

ance

bet

wee

n a

prei

mag

e an

d its

imag

e po

int

istw

ice

the

dist

ance

bet

wee

n th

e pa

ralle

l lin

es.

3.Il

lust

rate

you

r an

swer

s to

Exe

rcis

es 1

and

2 w

ith

sket

ches

.Use

ase

para

te s

heet

if n

eces

sary

.

See

stu

dent

s’w

ork.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-2

9-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Rot

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-3

9-3

©G

lenc

oe/M

cGra

w-H

ill49

1G

lenc

oe G

eom

etry

Lesson 9-3

Dra

w R

ota

tio

ns

A t

rans

form

atio

n ca

lled

a ro

tati

ontu

rns

a fi

gure

thr

ough

a s

peci

fied

angl

e ab

out

a fi

xed

poin

t ca

lled

the

cen

ter

of r

otat

ion

.To

find

the

imag

e of

a r

otat

ion,

one

way

is t

o us

e a

prot

ract

or.A

noth

er w

ay is

to

refl

ect

a fi

gure

tw

ice,

in t

wo

inte

rsec

ting

line

s.

!A

BC

has

ver

tice

s A

(2,1

),B

(3,4

),an

d

C(5

,1).

Dra

w t

he

imag

e of

!A

BC

un

der

a r

otat

ion

of

90°

cou

nte

rclo

ckw

ise

abou

t th

e or

igin

.•

Fir

st d

raw

!A

BC

.The

n dr

aw a

seg

men

t fr

om O

,the

ori

gin,

to p

oint

A.

•U

se a

pro

trac

tor

to m

easu

re 9

0°co

unte

rclo

ckw

ise

wit

h O !

A!as

one

sid

e.•

Dra

w O

R! !

" .•

Use

a c

ompa

ss t

o co

py O !

A!on

to O

R!!

" .N

ame

the

segm

ent

O!A!

!!.•

Rep

eat

wit

h se

gmen

ts f

rom

the

ori

gin

to p

oint

s B

and

C.

Fin

d t

he

imag

e of

!A

BC

un

der

ref

lect

ion

in

lin

es m

and

n.

Fir

st r

efle

ct !

AB

Cin

line

m.L

abel

the

imag

e !

A!B

!C!.

Ref

lect

!A!

B!C

!in

line

n.L

abel

the

imag

e !

A%B

%C%.

!A

%B%C

%is

a r

otat

ion

of !

AB

C.T

he c

ente

r of

rot

atio

n is

the

inte

rsec

tion

of

lines

man

d n.

The

ang

le o

f ro

tati

on is

tw

ice

the

mea

sure

of

the

acut

e an

gle

form

ed b

y m

and

n.

Dra

w t

he

rota

tion

im

age

of e

ach

fig

ure

90°

in t

he

give

n d

irec

tion

abo

ut

the

cen

ter

poi

nt

and

lab

el t

he

coor

din

ates

.

1.P!Q!

wit

h en

dpoi

nts

P("

1,"

2)

2.!

PQ

Rw

ith

vert

ices

P("

2,"

3),Q

(2,"

1),

and

Q(1

,3)

coun

terc

lock

wis

e an

d R

(3,2

) cl

ockw

ise

abou

t th

e po

int

T(1

,1)

abou

t th

e or

igin

Fin

d t

he

rota

tion

im

age

of e

ach

fig

ure

by

refl

ecti

ng

it i

n l

ine

man

d t

hen

in

lin

e n

.

3.4.

mn

B!

C!

A!

AB

C

B&

C&

A&

mn

Q!

P!

Q

PQ

&

P&

x

yP

!

Q!

R!

P

Q

RT

x

y

P!

Q!

P

Q

m

n

A! B

!C

! A&

B&

C&

B

C

A

x

y

O

A!

B!

C!

A

B

C

R

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill49

2G

lenc

oe G

eom

etry

Ro

tati

on

al S

ymm

etry

Whe

n th

e fi

gure

at

the

righ

t is

rot

ated

ab

out

poin

t P

by 1

20°

or 2

40°,

the

imag

e lo

oks

like

the

prei

mag

e.T

he

figu

re h

as r

otat

ion

al s

ymm

etry

,whi

ch m

eans

it c

an b

e ro

tate

d le

ss

than

360

°ab

out

a po

int

and

the

prei

mag

e an

d im

age

appe

ar t

o be

th

e sa

me.

The

fig

ure

has

rota

tion

al s

ymm

etry

of o

rder

3 be

caus

e th

ere

are

3 ro

tati

ons

less

tha

n 36

0°(0

°,12

0°,2

40°)

tha

t pr

oduc

e an

imag

e th

at

is t

he s

ame

as t

he o

rigi

nal.

The

mag

nit

ud

eof

the

rot

atio

nal s

ymm

etry

fo

r a

figu

re is

360

deg

rees

div

ided

by

the

orde

r.Fo

r th

e fi

gure

abo

ve,

the

rota

tion

al s

ymm

etry

has

mag

nitu

de 1

20 d

egre

es.

Iden

tify

th

e or

der

an

d m

agn

itu

de

of t

he

rota

tion

al

sym

met

ry o

f th

e d

esig

n a

t th

e ri

ght.

The

des

ign

has

rota

tion

al s

ymm

etry

abo

ut t

he c

ente

r po

int

for

rota

tion

s of

0°,

45°,

90°,

135°

,180

°,22

5°,2

70°,

and

315°

.

The

re a

re e

ight

rot

atio

ns le

ss t

han

360

degr

ees,

so t

he o

rder

of

its

rota

tion

al s

ymm

etry

is 8

.The

quo

tien

t 36

0 &

8 is

45,

so t

he m

agni

tude

of

its

rota

tion

al s

ymm

etry

is 4

5 de

gree

s.

Iden

tify

th

e or

der

an

d m

agn

itu

de

of t

he

rota

tion

al s

ymm

etry

of

each

fig

ure

.

1.a

squa

re2.

a re

gula

r 40

-gon

orde

r:4;

mag

nitu

de:9

0°or

der:

40;m

agni

tude

:9°

3.4.

orde

r:2;

mag

nitu

de:1

80°

orde

r:5;

mag

nitu

de:7

2°

5.6.

orde

r:16

;mag

nitu

de:2

2.5°

orde

r:3;

mag

nitu

de:1

20°

P

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Rot

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-3

9-3

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Rot

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-3

9-3

©G

lenc

oe/M

cGra

w-H

ill49

3G

lenc

oe G

eom

etry

Lesson 9-3

Rot

ate

each

fig

ure

abo

ut

poi

nt

Ru

nd

er t

he

give

n a

ngl

e of

rot

atio

n a

nd

th

e gi

ven

dir

ecti

on.L

abel

th

e ve

rtic

es o

f th

e ro

tati

on i

mag

e.

1.90

°co

unte

rclo

ckw

ise

2.90

°cl

ockw

ise

CO

OR

DIN

ATE

GEO

MET

RYD

raw

th

e ro

tati

on i

mag

e of

eac

h f

igu

re 9

0°in

th

e gi

ven

dir

ecti

on a

bou

t th

e or

igin

an

d l

abel

th

e co

ord

inat

es.

3.!

ST

Ww

ith

vert

ices

S(2

,"1)

,T(5

,1),

4.!

DE

Fw

ith

vert

ices

D("

4,3)

,E(1

,2),

and

W(3

,3)

coun

terc

lock

wis

ean

d F

("3,

"3)

clo

ckw

ise

Use

a c

omp

osit

ion

of

refl

ecti

ons

to f

ind

th

e ro

tati

on i

mag

e w

ith

res

pec

t to

lin

es k

and

m.T

hen

fin

d t

he

angl

e of

rot

atio

n f

or e

ach

im

age.

5.6.

120°

cloc

kwis

e14

0°co

unte

rclo

ckw

ise

k mL!

M!

N! O

!

LM N

Ok m

A!

B!C!A

BC

E!D

!F

!

F

ED

x

y

OS

!

T!

W!

S

T

W

x

y

O

GH

KJ

R

G!

H!

J!K!

Q

PS

R

Q!

S!

P!

©G

lenc

oe/M

cGra

w-H

ill49

4G

lenc

oe G

eom

etry

Rot

ate

each

fig

ure

abo

ut

poi

nt

Ru

nd

er t

he

give

n a

ngl

e of

rot

atio

n a

nd

th

e gi

ven

dir

ecti

on.L

abel

th

e ve

rtic

es o

f th

e ro

tati

on i

mag

e.

1.80

°co

unte

rclo

ckw

ise

2.10

0°cl

ockw

ise

CO

OR

DIN

ATE

GEO

MET

RYD

raw

th

e ro

tati

on i

mag

e of

eac

h f

igu

re 9

0°in

th

e gi

ven

dir

ecti

on a

bou

t th

e ce

nte

r p

oin

t an

d l

abel

th

e co

ord

inat

es.

3.!

RS

Tw

ith

vert

ices

R("

3,3)

,S(2

,4),

4.!

HJK

wit

h ve

rtic

es H

(3,1

),J(

3,"

3),

and

T(1

,2)

cloc

kwis

e ab

out

the

and

K("

3,"

4) c

ount

ercl

ockw

ise

abou

t po

int

P(1

,0)

the

poin

t P

("1,

"1)

Use

a c

omp

osit

ion

of

refl

ecti

ons

to f

ind

th

e ro

tati

on i

mag

e w

ith

res

pec

t to

lin

es p

and

s.T

hen

fin

d t

he

angl

e of

rot

atio

n f

or e

ach

im

age.

5.6.

160°

coun

terc

lock

wis

e10

0°co

unte

rclo

ckw

ise

7.ST

EAM

BO

ATS

A p

addl

e w

heel

on

a st

eam

boat

is d

rive

n by

a s

team

eng

ine

and

mov

esfr

om o

ne p

addl

e to

the

nex

t to

pro

pel t

he b

oat

thro

ugh

the

wat

er.I

f a

padd

le w

heel

cons

ists

of

18 e

venl

y sp

aced

pad

dles

,ide

ntif

y th

e or

der

and

mag

nitu

de o

f it

s ro

tati

onal

sym

met

ry.

orde

r 18

and

mag

nitu

de 2

0°

ps

P!

R!

S!

T!

P

R

ST

ps

E!

F!

G!

EFG

K!

J!H

!

JH

K

x

y

OP

( –1,

–1)

S!

R!

T!

T

S

R

x

y

OP

( 1, 0

)

QU

TS

P

RQ!

S!T

!U

! P!

N

MP

R

M!

N!

P!

Pra

ctic

e (A

vera

ge)

Rot

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-3

9-3



Rea

din

g t

o L

earn

Math

emati

csR

otat

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-3

9-3

©G

lenc

oe/M

cGra

w-H

ill49

5G

lenc

oe G

eom

etry

Lesson 9-3

Pre-

Act

ivit

yH

ow d

o so

me

amu

sem

ent

rid

es i

llu

stra

te r

otat

ion

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 9-

3 at

the

top

of

page

476

in y

our

text

book

.

Wha

t ar

e tw

o w

ays

that

eac

h ca

r ro

tate

s?E

ach

car

spin

s ar

ound

its

own

cent

er a

nd e

ach

car

rota

tes

arou

nd a

poi

nt in

the

cen

ter

of t

he c

ircu

lar

trac

k.

Rea

din

g t

he

Less

on

1.L

ist

all o

f th

e fo

llow

ing

type

s of

tra

nsfo

rmat

ions

tha

t sa

tisf

y ea

ch d

escr

ipti

on:r

efle

ctio

n,tr

ansl

atio

n,ro

tati

on.

a.T

he t

rans

form

atio

n is

an

isom

etry

.re

flect

ion,

tran

slat

ion,

rota

tion

b.T

he t

rans

form

atio

n pr

eser

ves

the

orie

ntat

ion

of a

fig

ure.

tran

slat

ion,

rota

tion

c.T

he t

rans

form

atio

n is

the

com

posi

te o

f su

cces

sive

ref

lect

ions

ove

r tw

o in

ters

ecti

ng

lines

.ro

tatio

nd

.T

he t

rans

form

atio

n is

the

com

posi

te o

f su

cces

sive

ref

lect

ions

ove

r tw

o pa

ralle

l lin

es.

tran

slat

ion

e.A

spe

cifi

c tr

ansf

orm

atio

n is

def

ined

by

a fi

xed

poin

t an

d a

spec

ifie

d an

gle.

rota

tion

f.A

spe

cifi

c tr

ansf

orm

atio

n is

def

ined

by

a fi

xed

poin

t,a

fixe

d lin

e,or

a f

ixed

pla

ne.

refle

ctio

ng.

A s

peci

fic

tran

sfor

mat

ion

is d

efin

ed b

y (x

,y) →

(x$

a,x

$b)

,for

fix

ed v

alue

s of

a

and

b.tr

ansl

atio

nh

.T

he t

rans

form

atio

n is

als

o ca

lled

a sl

ide.

tran

slat

ion

i.T

he t

rans

form

atio

n is

als

o ca

lled

a fl

ip.

refle

ctio

nj.

The

tra

nsfo

rmat

ion

is a

lso

calle

d a

turn

.ro

tatio

n

2.D

eter

min

e th

e or

der

and

mag

nitu

de o

f th

e ro

tati

onal

sym

met

ry f

or e

ach

figu

re.

a.b.

orde

r 3;

mag

nitu

de 1

20°

orde

r 2;

mag

nitu

de 1

80°

c.d.

orde

r 5;

mag

nitu

de 7

2°or

der

8;m

agni

tude

45°

Hel

pin

g Y

ou

Rem

emb

er

3.W

hat

is a

n ea

sy w

ay t

o re

mem

ber

the

orde

r an

d m

agni

tude

of

the

rota

tion

al s

ymm

etry

of a

reg

ular

pol

ygon

?S

ampl

e an

swer

:The

ord

er is

the

sam

e as

the

num

ber

of s

ides

.To

find

the

mag

nitu

de,d

ivid

e 36

0 by

the

num

ber

of s

ides

.

©G

lenc

oe/M

cGra

w-H

ill49

6G

lenc

oe G

eom

etry

Find

ing

the

Cen

ter

of R

otat

ion

Supp

ose

you

are

told

tha

t !

X!Y

!Z!

is t

he

rota

tion

imag

e of

!X

YZ

,but

you

are

not

tol

dw

here

the

cen

ter

of r

otat

ion

is n

or t

he m

easu

re

of t

he a

ngle

of

rota

tion

.Can

you

fin

d th

em?

Yes,

you

can.

Con

nect

tw

o pa

irs

of c

orre

spon

ding

vert

ices

wit

h se

gmen

ts.I

n th

e fi

gure

,the

se

gmen

ts Y

Y!

and

ZZ

!ar

e us

ed.D

raw

the

perp

endi

cula

r bi

sect

ors,

"an

d m

,of

thes

ese

gmen

ts.T

he p

oint

Cw

here

"an

d m

inte

rsec

t is

the

cen

ter

of r

otat

ion.

1.H

ow c

an y

ou f

ind

the

mea

sure

of

the

angl

e of

rot

atio

n in

the

fig

ure

abov

e?D

raw

"X

CX

!("

YC

Y!

or "

ZCZ

!w

ould

al

so d

o) a

nd m

easu

re it

with

a p

rotr

acto

r.

Loc

ate

the

cen

ter

of r

otat

ion

for

th

e ro

tati

on t

hat

map

s W

XY

Zon

to W

!X!Y

!Z!.

Th

en f

ind

th

e m

easu

re o

f th

e an

gle

of r

otat

ion

.

2.3.

The

segm

ents

stu

dent

s ch

oose

to

bise

ct m

ay v

ary.

See

stu

dent

s’w

ork.

Z

Z!

W

W!

X!

X

Y

Y!

cent

er o

fro

tatio

n

angl

e of

rota

tion:

abou

t 110

Z!

Z

W

W!

X!

X

Y!

Y

cent

er o

fro

tatio

n

angl

e of

rota

tion:

abou

t 100

Z!

Z

C

X!

X

Y!

Y

m"

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-3

9-3


©G

lencoe/McG

raw-H

illA

11G

lencoe Geom

etry

Answers

Study Guide and InterventionTessellations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4


Less

on

9-4

Regular Tessellations A pattern that covers a plane with repeating copies of one ormore figures so that there are no overlapping or empty spaces is a tessellation. A regulartessellation uses only one type of regular polygon. In a tessellation, the sum of the measuresof the angles of the polygons surrounding a vertex is 360. If a regular polygon has an interiorangle that is a factor of 360, then the polygon will tessellate.

regular tessellation tessellation Copies of a regular hexagon Copies of a regular pentagon can form a tessellation. cannot form a tessellation.

Determine whether a regular 16-gon tessellates the plane. Explain.If m"1 is the measure of one interior angle of a regular polygon, then a formula for m"1

is m"1 # '180(n

n" 2)'. Use the formula with n # 16.

m"1 # '180(n

n" 2)'

# '180(1

166

" 2)'

# 157.5

The value 157.5 is not a factor of 360, so the 16-gon will not tessellate.

Determine whether each polygon tessellates the plane. If so, draw a sample figure.

1. scalene right triangle yes 2. isosceles trapezoid yes


3. square 4. 20-gonYes; the measure of each interior No; the measure of each interior angle is 90, and 90 is a factor angle is 162, and 162 is not a factor of 360. of 360.

5. septagon 6. 15-gonNo; the measure of each interior No; the measure of each interior angle is 128.6, and 128.6 is not a angle is 156, and 156 is not a factor factor of 360. of 360.

7. octagon 8. pentagonNo; the measure of each interior No; the measure of each interior angle is 135, and 135 is not a angle is 108, and 108 is not a factor factor of 360. of 360.

ExercisesExercises

ExampleExample


Tessellations with Specific Attributes A tessellation pattern can contain any typeof polygon. If the arrangement of shapes and angles at each vertex in the tessellation is thesame, the tessellation is uniform. A semi-regular tessellation is a uniform tessellationthat contains two or more regular polygons.

Determine whether a kite will tessellate the plane. If so, describe the tessellation as uniform, regular, semi-regular, or not uniform.A kite will tessellate the plane. At each vertex the sum of the measures is a $ b $ b $ c, which is 360. The tessellation is uniform.

Determine whether a semi-regular tessellation can be created from each set offigures. If so, sketch the tessellation. Assume that each figure has a side length of1 unit.

1. rhombus, equilateral triangle, 2. square and equilateral triangleand octagon

no yes


3. rectangle 4. hexagon and square

yes, uniform no

c' a'

b'

b'

c'

c'

a'

a'b'

b'

b'

c'a'

b'

b'

c'a'

b'

b'

c'a'

b'

b'

b'

c' a'

b'

b'


Tessellations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4

ExercisesExercises

ExampleExample

Answ

ers(Lesson 9-4)

©G

lencoe/McG

raw-H

illA

12G

lencoe Geom

etry

Skills PracticeTessellations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4


Less

on

9-4


1. 15-gon 2. 18-gonno, interior angle $ 156 no, interior angle $ 160

3. square 4. 20-gonyes, interior angle $ 90 no, interior angle $ 162


5. regular pentagons and equilateral trianglesno

6. regular dodecagons and equilateral trianglesyes

7. regular octagons and equilateral trianglesno


8. rhombus 9. isosceles trapezoid and square

yes; uniform no


10. 11.

yes; uniform, semi-regular yes; uniform



1. 22-gon 2. 40-gonno, interior angle ≈ 163.6 no, interior angle $ 171


3. regular pentagons and regular decagonsno (will not tessellate the plane)

4. regular dodecagons, regular hexagons, and squaresyes


5. kite 6. octagon and decagon

yes; uniform no


7. 8.

yes; uniform, semi-regular yes, not uniform

FLOOR TILES For Exercises 9 and 10, use the following information.Mr. Martinez chose the pattern of tile shown to retile his kitchen floor.

9. Determine whether the pattern is a tessellation. ExplainYes; since there are 2 squares and 3 triangles at each vertex, the sum of the angles at the vertices is 360°.

10. Is the pattern uniform, regular, or semi-regular?uniform and semi-regular

Practice (Average)

Tessellations

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4

Answ

ers(Lesson 9-4)


An

swer

s

Rea

din

g t

o L

earn

Math

emati

csTe

ssel

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-4

9-4

©G

lenc

oe/M

cGra

w-H

ill50

1G

lenc

oe G

eom

etry

Lesson 9-4

Pre-

Act

ivit

yH

ow a

re t

esse

llat

ion

s u

sed

in

art

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 9-

4 at

the

top

of

page

483

in y

our

text

book

.

•In

the

pat

tern

sho

wn

in t

he p

ictu

re in

you

r te

xtbo

ok,h

ow m

any

smal

leq

uila

tera

l tri

angl

es m

ake

up o

ne r

egul

ar h

exag

on?

6•

In t

his

patt

ern,

how

man

y fi

sh m

ake

up o

ne e

quila

tera

l tri

angl

e?1%

1 2%

Rea

din

g t

he

Less

on

1.U

nder

line

the

corr

ect

wor

d,ph

rase

,or

num

ber

to f

orm

a t

rue

stat

emen

t.a.

A t

esse

llati

on is

a p

atte

rn t

hat

cove

rs a

pla

ne w

ith

the

sam

e fi

gure

or

set

of f

igur

es s

oth

at t

here

are

no

(con

grue

nt a

ngle

s/ov

erla

ppin

g or

em

pty

spac

es/r

ight

ang

les)

.b.

A t

esse

llati

on t

hat

uses

onl

y on

e ty

pe o

f re

gula

r po

lygo

n is

cal

led

a(u

nifo

rm/r

egul

ar/s

emi-

regu

lar)

tes

sella

tion

.c.

The

sum

of

the

mea

sure

s of

the

ang

les

at a

ny v

erte

x in

any

tes

sella

tion

is(9

0/18

0/36

0).

d.A

tes

sella

tion

tha

t co

ntai

ns t

he s

ame

arra

ngem

ent

of s

hape

s an

d an

gles

at

ever

yve

rtex

is c

alle

d a

(uni

form

/reg

ular

/sem

i-re

gula

r) t

esse

llati

on.

e.In

a r

egul

ar t

esse

llati

on m

ade

up o

f he

xago

ns,t

here

are

(3/

4/6)

hex

agon

s m

eeti

ng a

tea

ch v

erte

x,an

d th

e m

easu

re o

f ea

ch o

f th

e an

gles

at

any

vert

ex is

(60

/90/

120)

.f.

A u

nifo

rm t

esse

llati

on f

orm

ed u

sing

tw

o or

mor

e re

gula

r po

lygo

ns is

cal

led

a(r

otat

iona

l/reg

ular

/sem

i-re

gula

r) t

esse

llati

on.

g.In

a r

egul

ar t

esse

llati

on m

ade

up o

f tr

iang

les,

ther

e ar

e (3

/4/6

) tr

iang

les

mee

ting

at

each

ver

tex,

and

the

mea

sure

of

each

of

the

angl

es a

t an

y ve

rtex

is (

30/6

0/12

0).

h.

If a

reg

ular

tes

sella

tion

is m

ade

up o

f qu

adri

late

rals

,all

of t

he q

uadr

ilate

rals

mus

t be

cong

ruen

t (r

ecta

ngle

s/pa

ralle

logr

ams/

squa

res/

trap

ezoi

ds).

2.W

rite

all

of t

he f

ollo

win

g w

ords

tha

t de

scri

be e

ach

tess

ella

tion

:uni

form

,non

-uni

form

,re

gula

r,se

mi-

regu

lar.

a.no

n-un

iform

b.un

iform

,sem

i-reg

ular

Hel

pin

g Y

ou

Rem

emb

er

3.O

ften

the

eve

ryda

y m

eani

ngs

of a

wor

d ca

n he

lp y

ou t

o re

mem

ber

its

mat

hem

atic

alm

eani

ng.L

ook

up u

nifo

rmin

you

r di

ctio

nary

.How

can

its

ever

yday

mea

ning

s he

lp y

outo

rem

embe

r th

e m

eani

ng o

f a

unif

orm

tess

ella

tion

?S

ampl

e an

swer

:Thr

ee s

imila

rm

eani

ngs

of u

nifo

rmar

e un

vary

ing,

cons

iste

nt,a

nd id

entic

al.I

n a

unifo

rm t

esse

llatio

n,th

e ar

rang

emen

t of

sha

pes

and

angl

es a

t ea

chve

rtex

is t

he s

ame.

Eac

h of

the

thr

ee e

very

day

mea

ning

s de

scri

bes

this

situ

atio

n.

©G

lenc

oe/M

cGra

w-H

ill50

2G

lenc

oe G

eom

etry

Poly

gona

l Num

bers

Cer

tain

num

bers

rel

ated

to

regu

lar

poly

gons

are

cal

led

pol

ygon

al n

um

bers

.The

cha

rtsh

ows

seve

ral t

rian

gula

r,sq

uare

,and

pen

tago

nal n

umbe

rs.T

he r

ank

of a

pol

ygon

num

ber

is t

he n

umbe

r of

dot

s on

eac

h “s

ide”

of t

he o

uter

pol

ygon

.For

exa

mpl

e,th

e pe

ntag

onal

num

ber

22 h

as a

ran

k of

4.

Poly

gona

l num

bers

can

be

desc

ribe

d w

ith

form

ulas

.For

exa

mpl

e,a

tria

ngul

ar n

umbe

r T

of r

ank

r ca

n be

des

crib

ed b

y T

#'r(

r2$

1)'

.

An

swer

eac

h q

ues

tion

.

1.D

raw

a d

iagr

am t

o fi

nd t

he2.

Dra

w a

dia

gram

to

find

the

tr

iang

ular

num

ber

of r

ank

5.15

pent

agon

al n

umbe

r of

ran

k 5.

35

3.W

rite

a f

orm

ula

for

a sq

uare

num

ber

4.W

rite

a f

orm

ula

for

a pe

ntag

onal

Sof

ran

k r.

num

ber

Pof

ran

k r.

S$

r2P

$%r(

3r2"

1)%

5.W

hat

is t

he r

ank

of t

he p

enta

gona

l6.

Lis

t th

e he

xago

nal n

umbe

rs f

or r

anks

num

ber

70?

71

to 5

.(H

int:

Dra

w a

dia

gram

.)1,

6,15

,28,

45

Ran

k 1

Tria

ngle

13

610

14

916

15

1222

Squ

are

Pen

tago

n

Ran

k 2

Ran

k 3

Ran

k 4

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-4

9-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Dila

tions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-5

9-5

©G

lenc

oe/M

cGra

w-H

ill50

3G

lenc

oe G

eom

etry

Lesson 9-5

Cla

ssif

y D

ilati

on

sA

dil

atio

nis

a t

rans

form

atio

n in

whi

ch t

he im

age

may

be

adi

ffer

ent

size

tha

n th

e pr

eim

age.

A d

ilati

on r

equi

res

a ce

nter

poi

nt a

nd a

sca

le f

acto

r,r.

Let r

repr

esen

t the

sca

le fa

ctor

of a

dila

tion.

If | r

| (1,

then

the

dila

tion

is a

n en

larg

emen

t.

If | r

| #1,

then

the

dila

tion

is a

con

grue

nce

tran

sfor

mat

ion.

If 0

)| r

| )1,

then

the

dila

tion

is a

red

uctio

n.

Dra

w t

he

dil

atio

n i

mag

e of

!

AB

Cw

ith

cen

ter

Oan

d r

$2.

Dra

w O

A! !

" ,O

B!!

" ,an

d O

C!!

" .L

abel

poi

nts

A!,

B!,

and

C!

so t

hat

OA!

#2(

OA

),O

B!

#2(

OB

),an

d O

C!

#2(

OC

).!

A!B

!C!

is a

dila

tion

of !

AB

C.

Dra

w t

he

dil

atio

n i

mag

e of

eac

h f

igu

re w

ith

cen

ter

Can

d t

he

give

n s

cale

fac

tor.

Des

crib

e ea

ch t

ran

sfor

mat

ion

as

an e

nla

rgem

ent,

con

gru

ence

,or

red

uct

ion

.

1.r

#2

enla

rgem

ent

2.r

#'1 2'

redu

ctio

n

3.r

#1

cong

ruen

ce4.

r#

3en

larg

emen

t

5.r

#'2 3'

redu

ctio

n6.

r#

1co

ngru

ence

KJ

HF

C

G

M C

P

R V

W

M!

P! R

! V!

W!

CS

TS

!

T!

C

W Y

XZ

C

ST

RS

!T

!

R!

CB

A A!

B!

O

B A

CO

B A

C

A!

B!

C!

Exer

cises

Exer

cises

Exam

ple

Exam

ple

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etry

Iden

tify

th

e Sc

ale

Fact

or

If y

ou k

now

cor

resp

ondi

ng m

easu

rem

ents

for

a p

reim

age

and

its

dila

tion

imag

e,yo

u ca

n fi

nd t

he s

cale

fac

tor.

Det

erm

ine

the

scal

e fa

ctor

for

th

e d

ilat

ion

of

X #Y#

to A#

B#.D

eter

min

e w

het

her

th

e d

ilat

ion

is

an e

nla

rgem

ent,

red

uct

ion

,or

con

gru

ence

tra

nsf

orm

ati

on.

scal

e fa

ctor

#' pr

i em imagae gele

ln eg nt gh th'

#'8 4

u un ni it ts s'

#2

The

sca

le f

acto

r is

gre

ater

tha

n 1,

so t

he d

ilati

on is

an

enla

rgem

ent.

Det

erm

ine

the

scal

e fa

ctor

for

eac

h d

ilat

ion

wit

h c

ente

r C

.Det

erm

ine

wh

eth

er t

he

dil

atio

n i

s an

en

larg

emen

t,re

du

ctio

n,o

r co

ngr

uen

ce t

ran

sfor

ma

tion

.

1.C

GH

Jis

a d

ilati

on im

age

of C

DE

F.

2.!

CK

Lis

a d

ilati

on im

age

of !

CK

L.

2;en

larg

emen

t1;

cong

ruen

ce

3.S

TU

VW

Xis

a d

ilati

on im

age

4.!

CP

Qis

a d

ilati

on im

age

of !

CY

Z.

of M

NO

PQ

R.

%1 2% ;re

duct

ion

%1 3% ;re

duct

ion

5.!

EF

Gis

a d

ilati

on im

age

of !

AB

C.

6.!

HJK

is a

dila

tion

imag

e of

!H

JK.

1.5;

enla

rgem

ent

1;co

ngru

ence

C

J

K

H

CB

DG

F

EA

C

Z

Y

P

QC

MN

O

PQ

RU

X

ST V

W

CKL

G CD

H

FE

J

A B

X Y

C

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Dila

tions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-5

9-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Dila

tions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-5

9-5

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etry

Lesson 9-5

Dra

w t

he

dil

atio

n i

mag

e of

eac

h f

igu

re w

ith

cen

ter

Can

d t

he

give

n s

cale

fac

tor.

1.r

#2

2.r

#'1 4'

Fin

d t

he

mea

sure

of

the

dil

atio

n i

mag

e M#

!#N#!#o

r of

th

e p

reim

age

M#N#

usi

ng

the

give

n s

cale

fac

tor.

3.M

N#

3,r

#3

4.M

!N!

#7,

r#

21

M!N

!$9

MN

$%1 3%

CO

OR

DIN

ATE

GEO

MET

RYF

ind

th

e im

age

of e

ach

pol

ygon

,giv

en t

he

vert

ices

,aft

era

dil

atio

n c

ente

red

at

the

orig

in w

ith

a s

cale

fac

tor

of 2

.Th

en g

rap

h a

dil

atio

n

cen

tere

d a

t th

e or

igin

wit

h a

sca

le f

acto

r of

%1 2% .

5.J(

2,4)

,K(4

,4),

P(3

,2)

6.D

("2,

0),G

(0,2

),F

(2,"

2)

Det

erm

ine

the

scal

e fa

ctor

for

eac

h d

ilat

ion

wit

h c

ente

r C

.Det

erm

ine

wh

eth

er t

he

dil

atio

n i

s an

en

larg

emen

t,re

du

ctio

n,o

r co

ngr

uen

ce t

ran

sfor

ma

tion

.Th

e d

ash

edfi

gure

is

the

dil

atio

n i

mag

e.

7.8.

4;en

larg

emen

t1;

cong

ruen

ce t

rans

form

atio

n

CC

G!

F!

D!

G&

F&

D&

x

y

O

J!K

!

P!

J&K

&

P&

x

y

O

CC

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ill50

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eom

etry

Dra

w t

he

dil

atio

n i

mag

e of

eac

h f

igu

re w

ith

cen

ter

Can

d t

he

give

n s

cale

fac

tor.

1.r

#'3 2'

2.r

#'2 3'

Fin

d t

he

mea

sure

of

the

dil

atio

n i

mag

e A#

!#T#!#o

r of

th

e p

reim

age

A#T#

usi

ng

the

give

nsc

ale

fact

or.

3.A

T#

15,r

#'3 5'

4.A

T#

30,r

#"

'1 6'5.

A!T

!#

12,r

#'4 3'

A!T

!$

9A

!T!

$5

AT$

9

CO

OR

DIN

ATE

GEO

MET

RYF

ind

th

e im

age

of e

ach

pol

ygon

,giv

en t

he

vert

ices

,aft

era

dil

atio

n c

ente

red

at

the

orig

in w

ith

a s

cale

fac

tor

of 2

.Th

en g

rap

h a

dil

atio

n

cen

tere

d a

t th

e or

igin

wit

h a

sca

le f

acto

r of

%1 2% .

6.A

(1,1

),C

(2,3

),D

(4,2

),E

(3,1

)7.

Q("

1,"

1),R

(0,2

),S

(2,1

)

Det

erm

ine

the

scal

e fa

ctor

for

eac

h d

ilat

ion

wit

h c

ente

r C

.Det

erm

ine

wh

eth

er t

he

dil

atio

n i

s an

en

larg

emen

t,re

du

ctio

n,o

r co

ngr

uen

ce t

ran

sfor

ma

tion

.Th

e d

otte

dfi

gure

is

the

dil

atio

n i

mag

e.

8.%1 3% ;

redu

ctio

n9.

2;en

larg

emen

t

10.P

HO

TOG

RA

PHY

Est

ebe

enla

rged

a 4

-inc

h by

6-i

nch

phot

ogra

ph b

y a

fact

or o

f '5 2' .

Wha

tar

e th

e ne

w d

imen

sion

s of

the

pho

togr

aph?

10 in

.by

15 in

.

CC

S!

Q!

R!

S&

Q&

R&

x

y

O

C!

D!

E!

A!

C&

D&

E&

A&

x

y

O

C

C

Pra

ctic

e (A

vera

ge)

Dila

tions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-5

9-5



Rea

din

g t

o L

earn

Math

emati

csD

ilatio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-5

9-5

©G

lenc

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cGra

w-H

ill50

7G

lenc

oe G

eom

etry

Lesson 9-5

Pre-

Act

ivit

yH

ow d

o yo

u u

se d

ilat

ion

s w

hen

you

use

a c

omp

ute

r?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 9-

5 at

the

top

of

page

490

in y

our

text

book

.

In a

ddit

ion

to t

he e

xam

ple

give

n in

you

r te

xtbo

ok,g

ive

two

ever

yday

exam

ples

of

scal

ing

an o

bjec

t,on

e th

at m

akes

the

obj

ect

larg

er a

nd a

noth

erth

at m

akes

it s

mal

ler.

Sam

ple

answ

er:l

arge

r:en

larg

ing

aph

otog

raph

;sm

alle

r:m

akin

g a

scal

e m

odel

of

a bu

ildin

gR

ead

ing

th

e Le

sso

n1.

Eac

h of

the

val

ues

of r

give

n be

low

rep

rese

nts

the

scal

e fa

ctor

for

a d

ilati

on.I

n ea

chca

se,d

eter

min

e w

heth

er t

he d

ilati

on is

an

enla

rgem

ent,

a re

duct

ion,

or a

con

grue

nce

tran

sfor

mat

ion.

a.r

#3

enla

rgem

ent

b.r

#0.

5re

duct

ion

c.r

#"

0.75

redu

ctio

nd.

r#

"1

cong

ruen

ce t

rans

form

atio

ne.

r#

'2 3're

duct

ion

f.r

#"

'3 2'en

larg

emen

tg.

r#

"1.

01en

larg

emen

th

.r#

0.99

9re

duct

ion

2.D

eter

min

e w

heth

er e

ach

sent

ence

is a

lway

s,so

met

imes

,or

neve

rtr

ue.I

f th

e se

nten

ce is

not

alw

ays

true

,exp

lain

why

.Fo

r ex

plan

atio

ns,s

ampl

e an

swer

s ar

e gi

ven.

a.A

dila

tion

req

uire

s a

cent

er p

oint

and

a s

cale

fac

tor.

alw

ays

b.A

dila

tion

cha

nges

the

siz

e of

a f

igur

e.S

omet

imes

;if

the

dila

tion

is a

cong

ruen

ce t

rans

form

atio

n,th

e si

ze o

f th

e fig

ure

is u

ncha

nged

.c.

A d

ilati

on c

hang

es t

he s

hape

of

a fi

gure

.N

ever

;all

dila

tions

pro

duce

sim

ilar

figur

es,a

nd s

imila

r fig

ures

hav

e th

e sa

me

shap

e.d.

The

imag

e of

a f

igur

e un

der

a di

lati

on li

es o

n th

e op

posi

te s

ide

of t

he c

ente

r fr

om t

hepr

eim

age.

Som

etim

es;t

his

is o

nly

true

whe

n th

e sc

ale

fact

or is

nega

tive.

e.A

sim

ilari

ty t

rans

form

atio

n is

a c

ongr

uenc

e tr

ansf

orm

atio

n.S

omet

imes

;thi

s is

true

onl

y w

hen

the

scal

e fa

ctor

is 1

or

"1.

f.T

he c

ente

r of

a d

ilati

on is

its

own

imag

e.al

way

s

g.A

dila

tion

is a

n is

omet

ry.

Som

etim

es;t

his

is t

rue

only

whe

n th

e di

latio

n is

a co

ngru

ence

tra

nsfo

rmat

ion.

h.

The

sca

le f

acto

r fo

r a

dila

tion

is a

pos

itiv

e nu

mbe

r.S

omet

imes

;the

sca

le f

acto

rca

n be

any

pos

itive

or

nega

tive

num

ber.

i.D

ilati

ons

prod

uce

sim

ilar

figu

res.

alw

ays

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o ex

plai

n it

to

som

eone

els

e.Su

ppos

e th

at y

our

clas

smat

e L

ydia

is h

avin

g tr

oubl

e un

ders

tand

ing

the

rela

tion

ship

bet

wee

n si

mil

arit

ytr

ansf

orm

atio

nsan

d co

ngru

ence

tra

nsfo

rmat

ions

.How

can

you

exp

lain

thi

s to

her

?S

ampl

e an

swer

:A c

ongr

uenc

e tr

ansf

orm

atio

n pr

eser

ves

the

size

and

shap

e of

a f

igur

e,th

at is

,the

imag

e is

con

grue

nt t

o th

e pr

eim

age.

Asi

mila

rity

tra

nsfo

rmat

ion

pres

erve

s th

e sh

ape

of a

fig

ure,

that

is,t

heim

age

is s

imila

r to

the

pre

imag

e.A

sim

ilari

ty t

rans

form

atio

n is

aco

ngru

ence

tra

nsfo

rmat

ion

if th

e sc

ale

fact

or is

1 o

r "

1.

©G

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w-H

ill50

8G

lenc

oe G

eom

etry

Sim

ilar

Cir

cles

You

may

be

surp

rise

d to

lear

n th

at t

wo

nonc

ongr

uent

cir

cles

tha

t lie

inth

e sa

me

plan

e an

d ha

ve n

o co

mm

on in

teri

or p

oint

s ca

n be

map

ped

one

onto

the

oth

er b

y m

ore

than

one

dila

tion

.

1.H

ere

is d

iagr

am t

hat

sugg

ests

one

way

to

m

ap a

sm

alle

r ci

rcle

ont

o a

larg

er o

ne u

sing

a

dila

tion

.The

cir

cles

are

giv

en.T

he li

nes

sugg

est

how

to

find

the

cen

ter

for

the

dila

tion

.Des

crib

e ho

w t

he c

ente

r is

fou

nd.

Use

seg

men

ts in

the

dia

gram

to

nam

e th

e sc

ale

fact

or.

Dra

w t

he t

wo

com

mon

ext

erna

lta

ngen

ts.T

he p

oint

whe

re t

hey

inte

rsec

tis

the

cen

ter

of a

dila

tion

that

map

s #

Aon

to #

A!;

scal

e fa

ctor

$%A A!M M

!%

.

2.H

ere

is a

noth

er p

air

of n

onco

ngru

ent

circ

les

wit

h no

com

mon

inte

rior

poi

nt.F

rom

Exe

rcis

e 1,

you

know

you

can

loca

te a

poi

nt

off

to t

he le

ft o

f th

e sm

alle

r ci

rcle

tha

t is

the

cen

ter

for

a di

lati

onm

appi

ng #

Con

to #

C!.

Fin

d an

othe

r ce

nter

for

ano

ther

dila

tion

that

map

s #

Con

to #

C!.

Mar

k an

d la

bel s

egm

ents

to

nam

e th

esc

ale

fact

or.

Find

the

inte

rsec

tion

of t

he t

wo

com

mon

inte

rnal

tan

gent

s;sc

ale

fact

or:"

%C C!X X!

%.

X

X!

C!

CO

A !

M!

M

AO

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-5

9-5



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Vect

ors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill50

9G

lenc

oe G

eom

etry

Lesson 9-6

Mag

nit

ud

e an

d D

irec

tio

nA

vec

tor

is a

dir

ecte

d se

gmen

t re

pres

enti

ng a

qua

ntit

y th

at h

as b

oth

mag

nit

ud

e,or

leng

th,

and

dir

ecti

on.F

or e

xam

ple,

the

spee

d an

d di

rect

ion

of a

n ai

rpla

ne c

an b

e re

pres

ente

d by

a v

ecto

r.In

sym

bols

,a v

ecto

r is

w

ritt

en a

s A

B!

,whe

re A

is t

he in

itia

l poi

nt a

nd B

is t

he e

ndpo

int,

or a

s v !

.

A v

ecto

r in

sta

nda

rd p

osit

ion

has

its

init

ial p

oint

at

(0,0

) an

d ca

n be

rep

rese

nted

by

the

orde

red

pair

for

poi

nt B

.The

vec

tor

at t

he r

ight

can

be

expr

esse

d as

v !#

"5,3

#.

You

can

use

the

Dis

tanc

e Fo

rmul

a to

fin

d th

e m

agni

tude

| A

B!

| of

a ve

ctor

.You

can

des

crib

e th

e di

rect

ion

of a

vec

tor

bym

easu

ring

the

ang

le t

hat

the

vect

or f

orm

s w

ith

the

posi

tive

x-ax

is o

r w

ith

any

othe

r ho

rizo

ntal

line

.

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

AB!

for

A(5

,2)

and

B(8

,7).

Fin

d th

e m

agni

tude

.| A

B| #

$(x

2"

!x 1

)2$

!(y

2"

!y 1

)2!

#$

(8 "

5!

)2$

(!

7 "

2!

)2 !#

$34!

or a

bout

5.8

uni

ts

To f

ind

the

dire

ctio

n,us

e th

e ta

ngen

t ra

tio.

tan

A#

'5 3'T

he ta

ngen

t rat

io is

opp

osite

ove

r ad

jace

nt.

m"

A%

59.0

Use

a c

alcu

lato

r.

The

mag

nitu

de o

f th

e ve

ctor

is a

bout

5.8

uni

ts a

nd it

s di

rect

ion

is 5

9°.

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

AB!

for

the

give

n c

oord

inat

es.R

oun

d t

o th

en

eare

st t

enth

.

1.A

(3,1

),B

("2,

3)2.

A(0

,0),

B("

2,1)

5.4;

158.

2°2.

2;15

3.4°

3.A

(0,1

),B

(3,5

)4.

A("

2,2)

,B(3

,1)

5;53

.1°

5.1;

348.

7°

5.A

(3,4

),B

(0,0

)6.

A(4

,2),

B(0

,3)

5;23

3.1°

4.1;

166.

0°

x

y

O

B( 8

,7)

A( 5

, 2)

x

y

O

B( 5

, 3)

A( 0

, 0)

V!

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

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cGra

w-H

ill51

0G

lenc

oe G

eom

etry

Tran

slat

ion

s w

ith

Vec

tors

Rec

all t

hat

the

tran

sfor

mat

ion

(a,b

) →(a

$2,

b"

3)re

pres

ents

a t

rans

lati

on r

ight

2 u

nits

and

dow

n 3

unit

s.T

he v

ecto

r "2

,"3#

is a

noth

er w

ay t

ode

scri

be t

hat

tran

slat

ion.

Als

o,tw

o ve

ctor

s ca

n be

add

ed:"

a,b#

$"c

,d##

"a$

c,b

$d

#.T

hesu

m o

f tw

o ve

ctor

s is

cal

led

the

resu

ltan

t.

Gra

ph

th

e im

age

of p

aral

lelo

gram

RS

TU

un

der

th

e tr

ansl

atio

n b

y th

e ve

ctor

s m !

$$3

,"1%

and

n !

$$"

2,"

4%.

Fin

d th

e su

m o

f th

e ve

ctor

s.m!

$n!

#"3

,"1#

$""

2,"

4##

"3 "

2,"

1 "

4##

"1,"

5#T

rans

late

eac

h ve

rtex

of

para

llelo

gram

RS

TU

righ

t 1

unit

an

d do

wn

5 un

its.

Gra

ph

th

e im

age

of e

ach

fig

ure

un

der

a t

ran

slat

ion

by

the

give

n v

ecto

r(s)

.

1.!

AB

Cw

ith

vert

ices

A("

1,2)

,B(0

,0),

2.A

BC

Dw

ith

vert

ices

A("

4,1)

,B("

2,3)

,an

d C

(2,3

);m !

#"2

,"3#

C(1

,1),

and

D("

1,"

1);n!

#"3

"3#

3.A

BC

Dw

ith

vert

ices

A("

3,3)

,B(1

,3),

C(1

,1),

and

D("

3,1)

;the

sum

of

p !#

""2,

1#an

dq!

#"5

,"4#

Giv

en m !

$$1

,"2%

and

n!$

$"3,

"4%

,rep

rese

nt

each

of

the

foll

owin

g as

a s

ingl

eve

ctor

.

4.m !

$n!

5.n!

"m!

$"2,

"6%

$"4,

"2%

x

y

D!

A!

B! C

!

DAB C

x

y

D!

A!

B!

C!

D

A

B

C

x

y A!

B!C

!

A

B

C

x

y

R!S

!

T!

U!

R

S

TU

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Vect

ors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-6

9-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Vect

ors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill51

1G

lenc

oe G

eom

etry

Lesson 9-6

Wri

te t

he

com

pon

ent

form

of

each

vec

tor.

1.2.

$3,4

%$5

,"5%

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

RS!

for

the

give

n c

oord

inat

es.R

oun

d t

o th

en

eare

st t

enth

.

3.R

(2,"

3),S

(4,9

)4.

R(0

,2),

S(3

,12)

2&37#

'12

.2,8

0.5°

&10

9#

'10

.4,7

3.3°

5.R

(5,4

),S

("3,

1)6.

R(1

,5),

S("

4,"

6)

&73#

'8.

5,20

0.6°

&14

6#

'12

.1,2

45.6

°

Gra

ph

th

e im

age

of e

ach

fig

ure

un

der

a t

ran

slat

ion

by

the

give

n v

ecto

r(s)

.

7.!

AB

Cw

ith

vert

ices

A("

4,3)

,B("

1,4)

,8.

trap

ezoi

d w

ith

vert

ices

T("

4,"

2),

C("

1,1)

;t!#

"4,"

3#R

("1,

"2)

,S("

2,"

3),Y

("3,

"3)

;a!#

"3,1

#an

d b!

#"2

,4#

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

each

res

ult

ant

for

the

give

n v

ecto

rs.

9.y !

#"7

,0#,

z!#

"0,6

#10

.b!#

"3,2

#,c!

#""

2,3#

&85#

'9.

2,40

.6°

&26#

'5.

1,78

.7°

S!

Y!

T!

R!

SY

TR

x

y

O

C!

B!

A!

CB

A

x

y

O

x

y

O

B( –

2, 2

)

D( 3

, –3)

x

y

OC

( 1, –

1)

E( 4

, 3)

©G

lenc

oe/M

cGra

w-H

ill51

2G

lenc

oe G

eom

etry

Wri

te t

he

com

pon

ent

form

of

each

vec

tor.

1.2.

$6,4

%$5

,"8%

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

FG!

for

the

give

n c

oord

inat

es.R

oun

d t

o th

en

eare

st t

enth

.

3.F

("8,

"5)

,G("

2,7)

4.F

("4,

1),G

(5,"

6)

6&5#

'13

.4,6

3.4°

&13

0#

'11

.4,3

22.1

°

Gra

ph

th

e im

age

of e

ach

fig

ure

un

der

a t

ran

slat

ion

by

the

give

n v

ecto

r(s)

.

5.!

QR

Tw

ith

vert

ices

Q("

1,1)

,R(1

,4),

6.tr

apez

oid

wit

h ve

rtic

es J

("4,

"1)

,T

(5,1

);s !

#""

2,"

5#K

(0,"

1),L

("1,

"3)

,M("

2,"

3);c!

#"5

,4#

and

d!#

""2,

1#

Fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

each

res

ult

ant

for

the

give

n v

ecto

rs.

7.a!

#""

6,4#

,b!#

"4,6

#8.

e!#

""4,

"5#

,f!#

""1,

3#

2&26#

'10

.2,1

01.3

°&

29#'

5.4,

201.

8°

AV

IATI

ON

For

Exe

rcis

es 9

an

d 1

0,u

se t

he

foll

owin

g in

form

atio

n.

A je

t be

gins

a fl

ight

alo

ng a

pat

h du

e no

rth

at 3

00 m

iles

per

hour

.A w

ind

is b

low

ing

due

wes

tat

30

mile

s pe

r ho

ur.

9.F

ind

the

resu

ltan

t ve

loci

ty o

f th

e pl

ane.

abou

t 30

1.5

mph

10.F

ind

the

resu

ltan

t di

rect

ion

of t

he p

lane

.ab

out

5.7°

wes

t of

due

nor

th

L!M

!

J!K

!

LM

JK

x

y

O

T!

R!

Q!

T

R

Qx

y

O

x

y

O

K( –

2, 4

)

L(3,

–4)

x

y

O

A( –

2, –

2)

B( 4

, 2)

Pra

ctic

e (A

vera

ge)

Vect

ors

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-6

9-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csVe

ctor

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill51

3G

lenc

oe G

eom

etry

Lesson 9-6

Pre-

Act

ivit

yH

ow d

o ve

ctor

s h

elp

a p

ilot

pla

n a

fli

ght?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 9-

6 at

the

top

of

page

498

in y

our

text

book

.

Why

do

pilo

ts o

ften

hea

d th

eir

plan

es in

a s

light

ly d

iffe

rent

dir

ecti

on f

rom

thei

r de

stin

atio

n?S

ampl

e an

swer

:to

com

pens

ate

for

the

effe

ct o

fth

e w

ind

so t

hat

the

resu

lt w

ill b

e th

at t

he p

lane

will

act

ually

fly in

the

cor

rect

dir

ectio

n

Rea

din

g t

he

Less

on

1.Su

pply

the

mis

sing

wor

ds o

r ph

rase

s to

com

plet

e th

e fo

llow

ing

sent

ence

s.a.

A

is a

dir

ecte

d se

gmen

t re

pres

enti

ng a

qua

ntit

y th

at h

as b

oth

mag

nitu

de a

nd d

irec

tion

.b.

The

leng

th o

f a

vect

or is

cal

led

its

.c.

Tw

o ve

ctor

s ar

e pa

ralle

l if

and

only

if t

hey

have

the

sam

e or

di

rect

ion.

d.A

vec

tor

is in

if

it is

dra

wn

wit

h in

itia

l poi

nt a

t th

e or

igin

.e.

Tw

o ve

ctor

s ar

e eq

ual i

f an

d on

ly if

the

y ha

ve t

he s

ame

and

the

sam

e.

f.T

he s

um o

f tw

o ve

ctor

s is

cal

led

the

.g.

A v

ecto

r is

wri

tten

in

if it

is e

xpre

ssed

as

an o

rder

ed p

air.

h.

The

pro

cess

of

mul

tipl

ying

a v

ecto

r by

a c

onst

ant

is c

alle

d .

2.W

rite

eac

h ve

ctor

des

crib

ed b

elow

in c

ompo

nent

for

m.

a.a

vect

or in

sta

ndar

d po

siti

on w

ith

endp

oint

(a,b

)$a

,b%

b.a

vect

or w

ith

init

ial p

oint

(a,b

) an

d en

dpoi

nt (c

,d)

$c"

a,d

"b%

c.a

vect

or in

sta

ndar

d po

siti

on w

ith

endp

oint

("3,

5)$"

3,5%

d.a

vect

or w

ith

init

ial p

oint

(2,

"3)

and

end

poin

t (6

,"8)

$4,"

5%e.

a!$

b!if

a!#

""3,

5#an

d b!

#"6

,"4#

$3,1

%f.

5u!if

u!#

"8,"

6#$4

0,"

30%

g."

'1 3' v!if

v!#

""15

,24#

$5,"

8%h

.0.

5u!$

1.5v!

if u!

#"1

0,"

10#a

nd v!

#""

8,6#

$"7,

4%

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is t

o re

late

it t

o a

term

you

alr

eady

know

.You

lear

ned

abou

t sc

ale

fact

ors

whe

n yo

u st

udie

d si

mila

rity

and

dila

tion

s.H

ow is

the

idea

of a

sca

lar

rela

ted

to s

cale

fact

ors?

Sam

ple

answ

er:A

sca

lar

is th

e te

rm u

sed

for

a co

nsta

nt (a

spe

cific

rea

l num

ber)

whe

n w

orki

ng w

ith v

ecto

rs.A

vec

tor

has

both

mag

nitu

de a

nd d

irec

tion,

whi

le a

sca

lar

is ju

st a

mag

nitu

de.

Mul

tiply

ing

a ve

ctor

by

a po

sitiv

e sc

alar

cha

nges

the

mag

nitu

de o

f th

eve

ctor

,but

not

the

dir

ectio

n,so

it r

epre

sent

s a

chan

ge in

sca

le.

scal

ar m

ultip

licat

ion

com

pone

nt fo

rmresu

ltant

dire

ctio

nm

agni

tude

stan

dard

pos

ition

oppo

site

mag

nitu

de

vect

or

©G

lenc

oe/M

cGra

w-H

ill51

4G

lenc

oe G

eom

etry

Rea

ding

Mat

hem

atic

sM

any

quan

titi

es in

nat

ure

can

be t

houg

ht o

f as

vec

tors

.The

sci

ence

of

phys

ics

invo

lves

man

y ve

ctor

qua

ntit

ies.

In r

eadi

ng a

bout

app

licat

ions

of m

athe

mat

ics,

ask

your

self

whe

ther

the

qua

ntit

ies

invo

lve

only

mag

nitu

de o

r bo

th m

agni

tude

and

dir

ecti

on.T

he f

irst

kin

d of

qua

ntit

yis

cal

led

scal

ar.T

he s

econ

d ki

nd is

a v

ecto

r.

Cla

ssif

y ea

ch o

f th

e fo

llow

ing.

Wri

te s

cala

ror

vec

tor.

1.th

e m

ass

of a

boo

ksc

alar

2.a

car

trav

elin

g no

rth

at 5

5 m

phve

ctor

3.a

ballo

on r

isin

g 24

fee

t pe

r m

inut

eve

ctor

4.th

e si

ze o

f a

shoe

scal

ar

5.a

room

tem

pera

ture

of

22 d

egre

es C

elsi

ussc

alar

6.a

wes

t w

ind

of 1

5 m

phve

ctor

7.th

e ba

ttin

g av

erag

e of

a b

aseb

all p

laye

rsc

alar

8.a

car

trav

elin

g 60

mph

vect

or

9.a

rock

fal

ling

at 1

0 m

phve

ctor

10.y

our

age

scal

ar

11.t

he f

orce

of

Ear

th’s

gra

vity

act

ing

on a

mov

ing

sate

llite

vect

or

12.t

he a

rea

of a

rec

ord

rota

ting

on

a tu

rnta

ble

scal

ar

13.t

he le

ngth

of

a ve

ctor

in t

he c

oord

inat

e pl

ane

scal

ar

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-6

9-6



Stu

dy

Gu

ide

and I

nte

rven

tion

Tran

sfor

mat

ions

with

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-7

9-7

©G

lenc

oe/M

cGra

w-H

ill51

5G

lenc

oe G

eom

etry

Lesson 9-7

Tran

slat

ion

s an

d D

ilati

on

sA

vec

tor

can

be r

epre

sent

ed b

y th

e or

dere

d pa

ir "

x,y#

or

by t

he c

olu

mn

mat

rix

.Whe

n th

e or

dere

d pa

irs

for

all t

he v

erti

ces

of a

pol

ygon

are

plac

ed t

oget

her,

the

resu

ltin

g m

atri

x is

cal

led

the

vert

ex m

atri

xfo

r th

e po

lygo

n.

For

!A

BC

wit

h A

("2,

2),B

(2,1

),an

d C

("1,

"1)

,the

ver

tex

mat

rix

for

the

tria

ngle

is

.

For

!A

BC

abov

e,u

se a

mat

rix

to f

ind

th

e co

ord

inat

es o

f th

eve

rtic

es o

f th

e im

age

of !

AB

Cu

nd

er t

he

tran

slat

ion

(x,

y) →

(x#

3,y

"1)

.To

tra

nsla

te t

he f

igur

e 3

unit

s to

the

rig

ht,a

dd 3

to

each

x-c

oord

inat

e.To

tra

nsla

te t

hefi

gure

1 u

nit

dow

n,ad

d "

1 to

eac

h y-

coor

dina

te.

Ver

tex

Mat

rix

Tra

nsla

tion

V

erte

x M

atri

xof

!A

BC

Mat

rix

of !

A!B

!C!

$#

The

coo

rdin

ates

are

A!(

1,1)

,B!(

5,0)

,and

C!(

2,"

2).

For

!A

BC

abov

e,u

se a

mat

rix

to f

ind

th

e co

ord

inat

es o

f th

e ve

rtic

esof

th

e im

age

of !

AB

Cfo

r a

dil

atio

n c

ente

red

at

the

orig

in w

ith

sca

le f

acto

r 3.

Scal

eV

erte

x M

atri

xV

erte

x M

atri

xFa

ctor

of !

AB

Cof

!A

!B!C

!

3 *

#

The

coo

rdin

ates

are

A!(

"6,

6),B

!(6,

3),a

nd C

!("

3,"

3).

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n t

ran

slat

ion

s or

dil

atio

ns.

1.!

AB

Cw

ith

A(3

,1),

B("

2,4)

,C("

2,"

1);(

x,y)

→(x

"1,

y$

2)A

!(2,

3),B

!("

3,6)

,C!(

"3,

1)2.

para

llelo

gram

RS

TU

wit

h R

("4,

"2)

,S("

3,1)

,T(3

,4),

U(2

,1);

(x,y

) →(x

"4,

y"

3)

R!(

"8,

"5)

,S!(

"7,

"2)

,T!(

"1,

1),U

!("

2,"

2)3.

rect

angl

e P

QR

Sw

ith

P(4

,0),

Q(3

,"3)

,R("

3,"

1),S

("2,

2);(

x,y)

→(x

"2,

y$

1)P

!(2,

1),Q

!(1,

"2)

,R!(

"5,

0),S

!("

4,3)

4.!

AB

Cw

ith

A("

2,"

1),B

("2,

"3)

,C(2

,"1)

;dila

tion

cen

tere

d at

the

ori

gin

wit

h sc

ale

fact

or 2

A!(

"4,

"2)

,B!(

"4,

"6)

,C!(

4,"

2)5.

para

llelo

gram

RS

TU

wit

h R

(4,"

2),S

("4,

"1)

,T("

2,3)

,U(6

,2);

dila

tion

cen

tere

d at

the

orig

in w

ith

scal

e fa

ctor

1.5

R!(

6,"

3),S

!("

6,"

1.5)

,T!(

"3,

4.5)

,U!(

9,3)

⎡"6

6"

3 ⎤⎣

63

"3⎦

⎡"2

2"

1 ⎤⎣

21

"1⎦

⎡15

2 ⎤⎣ 1

0"

2⎦⎡

33

3 ⎤⎣ "

1"

1"

1⎦⎡"

22

"1 ⎤

⎣2

1"

1⎦

⎡"2

2"

1 ⎤⎣

21

"1⎦

x

yB

( 2, 1

)

C( –

1, –

1)

A( –

2, 2

)

⎡x⎤

⎣ y⎦

Exer

cises

Exer

cises

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

©G

lenc

oe/M

cGra

w-H

ill51

6G

lenc

oe G

eom

etry

Ref

lect

ion

s an

d R

ota

tio

ns

Whe

n yo

u re

flec

t an

imag

e,on

e w

ay t

o fi

nd t

heco

ordi

nate

s of

the

ref

lect

ed v

erti

ces

is t

o m

ulti

ply

the

vert

ex m

atri

x of

the

obj

ect

by a

refl

ecti

on m

atri

x.To

per

form

mor

e th

an o

ne r

efle

ctio

n,m

ulti

ply

by o

ne r

efle

ctio

n m

atri

xto

fin

d th

e fi

rst

imag

e.T

hen

mul

tipl

y by

the

sec

ond

mat

rix

to f

ind

the

fina

l im

age.

The

mat

rice

s fo

r re

flec

tion

s in

the

axe

s,th

e or

igin

,and

the

line

y#

xar

e sh

own

belo

w.

For

a re

flect

ion

in t

he:

x-ax

isy-

axis

orig

inlin

e y

$x

Mul

tiply

the

ver

tex

mat

rix

by:

!A

BC

has

ver

tice

s A

("2,

3),B

(1,4

),an

d

C(3

,0).

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

th

e im

age

of !

AB

Caf

ter

a re

flec

tion

in

th

e x-

axis

.To

ref

lect

in t

he x

-axi

s,m

ulti

ply

the

vert

ex m

atri

x of

!A

BC

by t

here

flec

tion

mat

rix

for

the

x-ax

is.

Ref

lect

ion

Mat

rix

Ver

tex

Mat

rix

Ver

tex

Mat

rix

for

x-ax

isof

!A

BC

of !

A’B

’C’

*#

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

efle

ctio

n.

1.!

AB

Cw

ith

A("

3,2)

,B("

1,3)

,C(1

,0);

refl

ecti

on in

the

x-a

xis

A!(

"3,

"2)

,B!(

"1,

"3)

,C!(

1,0)

2.!

XY

Zw

ith

X(2

,"1)

,Y(4

,"3)

,Z("

2,1)

;ref

lect

ion

in t

he y

-axi

sX

!("

2,"

1),Y

!("

4,"

3),Z

!(2,

1)

3.!

AB

Cw

ith

A(3

,4),

B("

1,0)

,C("

2,4)

;ref

lect

ion

in t

he o

rigi

n A

!("

3,"

4),B

!(1,

0),C

!(2,

"4)

4.pa

ralle

logr

am R

ST

Uw

ith

R("

3,2)

,S(3

,2),

T(5

,"1)

,U("

1,"

1);r

efle

ctio

n in

the

line

y#

xR

!(2,

"3)

,S!(

2,3)

,T!(

"1,

5),U

!("

1,"

1)

5.!

AB

Cw

ith

A(2

,3),

B("

1,2)

,C(1

,"1)

;ref

lect

ion

in t

he o

rigi

n,th

en r

efle

ctio

n in

the

line

y#

xA

!("

3,"

2),B

!("

2,1)

,C!(

1,"

1)

6.pa

ralle

logr

am R

ST

Uw

ith

R(0

,2),

S(4

,2),

T(3

,"2)

,U("

1,"

2);r

efle

ctio

n in

the

x-a

xis,

then

ref

lect

ion

in t

he y

-axi

s R

!(0,

"2)

,S!(

"4,

"2)

,T!(

"3,

2),U

!(1,

2)

⎡"2

13 ⎤

⎣ "3

"4

0⎦⎡"

21

3 ⎤⎣

34

0⎦⎡1

0 ⎤⎣ 0

"1⎦

x

y

A!

B!

AB

C

⎡01⎤

⎣10⎦

⎡"1

0⎤⎣

0"

1⎦⎡"

10⎤

⎣0

1⎦⎡1

0⎤⎣0

"1⎦

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Tran

sfor

mat

ions

with

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-7

9-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Tran

sfor

mat

ions

with

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-7

9-7

©G

lenc

oe/M

cGra

w-H

ill51

7G

lenc

oe G

eom

etry

Lesson 9-7

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n t

ran

slat

ion

s.

1.!

ST

Uw

ith

S(6

,4),

T(9

,7),

and

U(1

4,2)

;(x,

y) →

(x"

4,y

$3)

S!(

2,7)

,T!(

5,10

),U

!(10

,5)

2.!

GH

Iw

ith

G("

5,0)

,H("

3,6)

,and

I("

2,1)

;(x,

y) →

(x$

2,y

$6)

G!(

"3,

6),H

!("

1,12

),I!

(0,7

)

Use

sca

lar

mu

ltip

lica

tion

to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

each

fig

ure

for

a d

ilat

ion

cen

tere

d a

t th

e or

igin

wit

h t

he

give

n s

cale

fac

tor.

3.!

DE

Fw

ith

D(2

,1),

E(5

,4),

and

F(7

,2);

r#

4D

!(8,

4),E

!(20

,16)

,F!(

28,8

)

4.qu

adri

late

ral W

XY

Zw

ith

W("

9,6)

,X("

6,3)

,Y(3

,12)

,and

Z("

6,15

);r

#'1 3'

W!(

"3,

2),X

!("

2,1)

,Y!(

1,4)

,Z!(

"2,

5)

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

efle

ctio

n.

5.!

MN

Ow

ith

M("

5,1)

,N("

2,3)

,and

O(2

,0);

y-ax

isM

!(5,

1),N

!(2,

3),O

!("

2,0)

6.qu

adri

late

ral A

BC

Dw

ith

A(3

,1),

B(6

,"2)

,C(5

,"5)

,and

D(1

,"6)

;x-a

xis

A!(

3,"

1),B

!(6,

2),C

!(5,

5),D

!(1,

6)

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

otat

ion

.

7.!

RS

Tw

ith

R("

2,"

2),S

("3,

3),a

nd T

(2,2

);90

°cou

nter

cloc

kwis

eR

!(2,

"2)

,S!(

"3,

"3)

,T!(

"2,

2)

8.$

LM

NP

wit

h L

(3,4

),M

(7,4

),N

(9,"

3),a

nd P

(5,"

3);1

80°

coun

terc

lock

wis

eL!

("3,

"4)

,M!(

"7,

"4)

,N!(

"9,

3),P

!("

5,3)

©G

lenc

oe/M

cGra

w-H

ill51

8G

lenc

oe G

eom

etry

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n t

ran

slat

ion

s.

1.!

KL

Mw

ith

K("

7,"

3),L

(4,9

),an

d M

(9,"

6);(

x,y)

→(x

"7,

y$

2)K

!("

14,"

1),L

!("

3,11

),M

!(2,

"4)

2.$

AB

CD

wit

h A

("4,

3),B

("2,

8),C

(3,1

0),a

nd D

(1,5

);(x

,y) →

(x$

3,y

"9)

A!(

"1,

"6)

,B!(

1,"

1),C

!(6,

1),D

!(4,

"4)

Use

sca

lar

mu

ltip

lica

tion

to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

each

fig

ure

for

a d

ilat

ion

cen

tere

d a

t th

e or

igin

wit

h t

he

give

n s

cale

fac

tor.

3.qu

adri

late

ral H

IJK

wit

h H

("2,

3),I

(2,6

),J(

8,3)

,and

K(3

,"4)

;r#

"'1 3'

H! !%2 3% ,

"1 ",

I!!"

%2 3% ,"

2 ",J

! !"%8 3% ,

"1 ",

K! !"

1,%4 3% "

4.pe

ntag

on D

EF

GH

wit

h D

("8,

"4)

,E("

8,2)

,F(2

,6),

G(8

,0),

and

H(4

,"6)

;r#

'5 4'

D!(

"10

,"5)

,E! !"

10,%

5 2% ",F

! !%5 2% ,%1 25 %

",G!(

10,0

),H

! !5,"

%1 25 %"

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

efle

ctio

n.

5.!

QR

Sw

ith

Q("

5,"

4),R

("1,

"1)

,and

S(2

,"6)

;x-a

xis

Q!(

"5,

4),R

!("

1,1)

,S!(

2,6)

6.qu

adri

late

ral V

XY

Zw

ith

V("

4,"

2),X

("3,

4),Y

(2,1

),an

d Z

(4,"

3);y

#x

V!(

"2,

"4)

,X!(

4,"

3),Y

!(1,

2),Z

!("

3,4)

Use

a m

atri

x to

fin

d t

he

coor

din

ates

of

the

vert

ices

of

the

imag

e of

eac

h f

igu

reu

nd

er t

he

give

n r

otat

ion

.

7.$

EF

GH

wit

h E

("5

"4)

,F("

3,"

1),G

(5,"

1),a

nd H

(3,"

4);9

0°co

unte

rclo

ckw

ise

E!(

4,"

5),F

!(1,

"3)

,G!(

1,5)

,H!(

4,3)

8.qu

adri

late

ral P

ST

Uw

ith

P("

3,5)

,S(2

,6),

T(8

,1),

and

U("

6,"

4);2

70°

coun

terc

lock

wis

eP

!(5,

3),S

!(6,

"2)

,T!(

1,"

8),U

!("

4,6)

9.FO

RES

TRY

A r

esea

rch

bota

nist

map

ped

a se

ctio

n of

for

este

d la

nd o

n a

coor

dina

te g

rid

to k

eep

trac

k of

end

ange

red

plan

ts in

the

reg

ion.

The

ver

tice

s of

the

map

are

A("

2,6)

,B

(9,8

),C

(14,

4),a

nd D

(1,"

1).A

fter

a m

onth

,the

bot

anis

t ha

s de

cide

d to

dec

reas

e th

e re

sear

ch a

rea

to '3 4'

of it

s or

igin

al s

ize.

If t

he c

ente

r fo

r th

e re

duct

ion

is O

(0,0

),wha

t ar

e th

e co

ordi

nate

s of

the

new

res

earc

h ar

ea?

A!"

%3 2% ,%9 2% ",

B!%2 47 %

,6",C

!%2 21 %,3

",D!%3 4% ,

"%3 4% "

Pra

ctic

e (A

vera

ge)

Tran

sfor

mat

ions

with

Mat

rice

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-7

9-7



Rea

din

g t

o L

earn

Math

emati

csTr

ansf

orm

atio

ns w

ith M

atri

ces

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-7

9-7

©G

lenc

oe/M

cGra

w-H

ill51

9G

lenc

oe G

eom

etry

Lesson 9-7

Pre-

Act

ivit

yH

ow c

an m

atri

ces

be u

sed

to

mak

e m

ovie

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 9-

7 at

the

top

of

page

506

in y

our

text

book

.

•W

hat

kind

of

tran

sfor

mat

ion

shou

ld b

e us

ed t

o m

ove

a po

lygo

n?re

flect

ion,

tran

slat

ion,

or r

otat

ion

•W

hat

kind

of t

rans

form

atio

n sh

ould

be

used

to

resi

ze a

pol

ygon

?di

latio

n

Rea

din

g t

he

Less

on

1.W

rite

a v

erte

x m

atri

x fo

r ea

ch f

igur

e.

a.!

AB

Cb.

para

llelo

gram

AB

CD

2.M

atch

eac

h tr

ansf

orm

atio

n fr

om t

he f

irst

col

umn

wit

h th

e co

rres

pond

ing

mat

rix

from

the

seco

nd o

r th

ird

colu

mn.

In e

ach

case

,the

ver

tex

mat

rix

for

the

prei

mag

e of

a f

igur

eis

mul

tipl

ied

on t

he le

ft b

y on

e of

the

mat

rice

s be

low

to

obta

in t

he im

age

of t

he f

igur

e.A

ll ro

tati

ons

liste

d ar

e co

unte

rclo

ckw

ise

thro

ugh

the

orig

in.(

Som

e m

atri

ces

may

be

used

mor

e th

an o

nce

or n

ot a

t al

l.)

a.re

flec

tion

ove

r th

e y-

axis

vii.

v.b.

90°

rota

tion

viii

c.re

flec

tion

ove

r th

e lin

e y

#x

iiiii

.vi

.d.

270°

rota

tion

ve.

refl

ecti

on o

ver

the

orig

inii

iii.

vii.

f.18

0°ro

tati

onii

g.re

flec

tion

ove

r th

e x-

axis

vii

iv.

viii

.h

.36

0°ro

tati

oni

Hel

pin

g Y

ou

Rem

emb

er3.

How

can

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rem

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r or

qui

ckly

fig

ure

out

the

mat

rice

s fo

r th

e tr

ansf

orm

atio

ns in

Exe

rcis

e 2?

Vis

ualiz

e or

ske

tch

how

the

“uni

t poi

nts”

(1,0

) on

the

x-ax

is a

nd(0

,1)

on t

he y

-axi

s ar

e m

oved

by

the

tran

sfor

mat

ion.

Wri

te t

he o

rder

ed p

air

for

the

imag

e po

ints

in a

2 (

2 m

atri

x,w

ith th

e co

ordi

nate

s of

the

imag

e of

the

x-ax

is u

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oint

in th

e fir

st c

olum

n an

d th

e co

ordi

nate

s of

the

imag

e of

the

y-ax

is u

nit

poin

t in

the

sec

ond

colu

mn.

⎡0"

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0⎦⎡

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eom

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Vect

or A

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onV

ecto

rs a

re p

hysi

cal q

uant

itie

s w

ith

mag

nitu

de a

nd d

irec

tion

.For

ce a

nd v

eloc

ity

are

two

exam

ples

.We

will

inve

stig

ate

addi

ng v

ecto

r qu

anti

ties

.The

sum

of

two

vect

ors

is c

alle

d a

resu

ltan

t ve

ctor

or ju

st t

he r

esul

tant

.

Tw

o se

par

ate

forc

es,o

ne

mea

suri

ng

20 u

nit

s an

d t

he

oth

er m

easu

rin

g 40

un

its,

act

on a

n

obje

ct.I

f th

e an

gle

betw

een

th

e fo

rces

is

50°,

fin

d t

he

mag

nit

ud

e an

d d

irec

tion

of

the

resu

ltan

t fo

rce.

Fir

st,t

he v

ecto

rs m

ust

be r

earr

ange

d by

pla

cing

the

tai

l of

the

20-u

nit

vect

or a

t th

e he

ad o

f th

e 40

-uni

t ve

ctor

.Sin

ce t

hese

ve

ctor

s ar

e no

t pe

rpen

dicu

lar,

the

hori

zont

al a

nd v

erti

cal

com

pone

nts

of o

ne o

f th

e ve

ctor

s m

ust

be f

ound

.Usi

ng

trig

onom

etry

,the

hor

izon

tal c

ompo

nent

mus

t be

(20

cos

50°

) un

its

and

the

vert

ical

com

pone

nt m

ust

be (

20 s

in 5

0°)

unit

s.R

epla

cing

the

20-

unit

vec

tor

wit

h th

ese

com

pone

nts,

we

can

now

for

m t

wo

vect

ors

perp

endi

cula

r an

d us

e th

e P

ytha

gore

an

The

orem

to

find

the

res

ulta

nt.

r2#

(40

$20

cos

50°

)2$

(20

sin

50°)

2

r2%

(52.

9)2

$(1

5.3)

2

r2%

3032

.5r2

%55

.1

tan

O#' 40

2 $02s 0in

c5 os0°50

°'

%0.

2898

m"

O%

16

The

refo

re,t

he r

esul

tant

for

ce is

55.

1 un

its

dire

cted

16°

from

the

40-

unit

for

ce.

Sol

ve.R

oun

d a

ll a

ngl

e m

easu

res

to t

he

nea

rest

deg

ree.

Rou

nd

all

oth

er m

easu

res

to t

he

nea

rest

ten

th.

1.A

pla

ne f

lies

due

wes

t at

250

kilo

met

ers

per

hour

whi

le t

he w

ind

blow

s so

uth

at 7

0ki

lom

eter

s pe

r ho

ur.F

ind

the

plan

e’s

resu

ltan

t ve

loci

ty.

259.

6 km

/h,1

6'so

uth

of w

est

2.A

pla

ne f

lies

east

for

200

km

,the

n 60

°so

uth

of e

ast

for

80 k

m.F

ind

the

plan

e’s

dist

ance

and

dire

ctio

n fr

om it

s st

arti

ng p

oint

.

249.

8 km

,16'

sout

h of

eas

t

3.O

ne f

orce

of

100

unit

s ac

ts o

n an

obj

ect.

Ano

ther

for

ce o

f 80

uni

ts a

cts

on t

he o

bjec

t at

a40

°an

gle

from

the

fir

st f

orce

.Fin

d th

e re

sult

ant

forc

e on

the

obj

ect.

169.

3 un

its,1

8'fr

om t

he 1

00-u

nit

forc

e

O

20

40 #

20

cos

50'

20 s

in 5

0'r

20

20 c

os 5

0'

20 s

in 5

0'

50'

O

20 u

nits

40 u

nits

50'

O

20 u

nits

40 u

nits

50'

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

9-7

9-7

Exam

ple

Exam

ple


chapter 9 resource masters - math class · 2019-11-24 · ©glencoe/mcgraw-hill iv glencoe geometry...

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