Name Class Date 9-1 - Pequannock Township High Schoolhs.pequannock.org/ourpages/auto/2014/8/26/59171599... · Name Class Date 9-1 Additional Vocabulary Support Translations Choose

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1

Name Class Date

9-1 Additional Vocabulary SupportTranslations

Choose the word from the list below that best matches each phrase.

composition of transformations corresponding parts image

rigid motion preimage translation

transformation(s)

1. the fi gure that results from a transformation

2. the original fi gure in a transformation

3. fl ipping, sliding, or turning a fi gure

4. two or more transformations in combination

Use a word from the list above to complete each sentence.

5. Th is transformation is an example of a because the fi gure slides in one direction, but does not fl ip, turn, or change size.

6. Th is translation is an example of a(n) because it preserves distance and angle measures.

7. In a translation, the sides or angles of the preimage and image that have the same lengths or angle measures are .

Multiple Choice

8. For the transformation shown at the right, triangle ABC is called the

corresponding part. image.

composition. preimage.

9. What type of transformation is shown at the right?

a fl ip a slide

a reduction a turn

46 2 4 6

2

4

6

4

6

x

yA

A

B

CB

C

translation

rigid motion

corresponding parts

image

preimage

transformation

composition of transformations

D

H


2

Name Class Date

9-1 Think About a Plan Translations

Coordinate Geometry Quadrilateral PLAT has vertices P(22, 0), L(21, 1), A(0, 1), and T(21, 0). T,2, 23.(PLAT) 5 P rLrArT r. Show that PPr, LLr, AAr, and TTr are all parallel.

Understanding the Problem

1. How can you use the three sentences of the problem to help you break the problem down into manageable pieces?

Planning the Solution

2. How can you use a diagram to help you solve the problem?

3. Use the given translation to fi nd the image points.

4. Graph PPr, LLr, AAr, and TT r. What theorems about parallel lines come to mind when you look at the graph?

Getting an Answer

5. Find the slopes of the lines that connect each original point to its image.

6. How can you use the slopes to establish that PPr, LLr, AAr, and TT r are all parallel?

Answers may vary. Sample: The fi rst sentence contains the given

information. The second gives the translation rule I can use to obtain the

coordinates of the image. The third directs me toward what I need to prove.

Plot the given points and the points of the image

on a coordinate grid.

If the slopes of two lines are equal, then the lines are parallel.

The lines are parallel because the lines all have the same slope.

P9(0, 23), L9(1, 22), A9(2, 22), T9(1, 23)

2

3

4

2 1 1 2

y

x

P

P

T

T

A

A

L

L

232; 23

2; 232; 23

2

Prentice Hall Gold Geometry • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

3

Name Class Date

9-1 Practice Form G

Translations

Tell whether the transformation appears to be a rigid motion. Explain.

1. 2.

3. 4.

In each diagram, the dashed-line fi gure is an image of the solid-line fi gure. (a) Choose an angle or point from the preimage and name its image. (b) List all pairs of corresponding sides.

5. 6.

Graph the image of each fi gure under the given translation.

7. T,21, 4.(nABC) 8. T,3, 3.(MNOP)

Th e dashed-line fi gure is a translation image of the solid-line fi gure. Write a rule to describe each translation.

9. 10.

Preimage Image Preimage Image

Preimage Image Preimage Image

B

AA

C

B

C

U

X

V

W

J K

LM

2 41

2

2

x

y

A

BC P O

NM

3 5

3

5

x

y

D C

A

A B

D CB

33

3

x

y

X

WV

U

Y

XW

VU

Y2

2

1

4y

A

BC P O

NM

yes; preserves distance and angle measures no; does not preserve distance

yes; preserves distance and angle measures yes; preserves distance and angle measures

(a) Sample: A9 is image of A. (b) AB and A9B9, BC and

B9C9, CA and C9A9

(a) Sample: M is image of V. (b) VW and MJ, WX and JK,

XU and KL, UV and LM

TR23, 24S(ABCD) T

R4, 23S(UVWXY)


4

Name Class Date

11. You are visiting Washington, D.C. From the American History Museum you walk 5 blocks east and 1 block south to the Air and Space Museum. Th en you walk 8 blocks west to the Washington Monument. Where is the Washington Monument in relation to the American History Museum?

12. You and some friends go to a book fair where booths are set out in rows. You buy drinks at the refreshment stand and then walk 8 rows north and 2 rows east to the science fi ction booth. Th en you walk 1 row south and 2 rows west to the children’s book booth. Where is the children’s book booth in relation to the refreshment stand?

13. Reasoning If T,10, 15.(PQRS) 5 PrQrRrSr, what translation maps PrQrRrSr onto PQRS?

14. nXYZ has coordinates X(2, 3), Y(1, 4), and Z(8, 9). A translation maps X to Xr(4, 7). What are the coordinates for Yr and Zr for this translation?

15. Use the graph at the right. Write three diff erent translation rules for which the image of nRST has a vertex at the origin.

16. Use the graph at the right. Write three diff erent translation rules for which the image of nBCD has a vertex at the origin.

Graph the image of each fi gure under the given translation.

17. T,23, 4.(nDEF) 18. T,25, 1.(KLMN)

R

S

T

5

2

2

4y

x

D

C

B

3 3

3

3y

x

D E

F2

2 2

2

4

y

x

2

2

2

4

y

x

K

N M

L

9-1 Practice (continued) Form G

Translations

TR210, 215S(P9Q9R9S9) 5 PQRS

Y9(3, 8); Z9(10, 13)

TR3, 23S(kBCD); T

R24, 21S(kBCD); TR0, 2S(kBCD)

TR21, 23S(kRST ); T

R25, 21S(kRST ); TR22, 2S(kRST )

1 block south and 3 blocks west

7 rows north

D E

F

K

N M

L

Prentice Hall Foundations Geometry • Teaching ResourcesCopyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.

5

Name Class Date

9-1 Practice Form K

Translations


1. 2.

In each diagram, the dashed-line fi gure is an image of the solid-line fi gure. (a) Choose an angle or point from the preimage and name its image. (b) List all pairs of corresponding sides.

3. 4.

Copy each graph. Graph the image of each fi gure under the given translation.

5. T,3, 24.(MATH) Describe in words the translation to the right 3 units and down 4 units.

To start, identify the coordinates of each vertex.

Th e vertices are:

M(u, 1), A(u, 3), T(22, u), and H(21, u).

6. T,22, 3.(nABC) 7. T,2, 23.(nXYZ)

Preimage

Image

Preimage Image

P

R

QP

Q

R

U UT T

S SR R

M

A T

H x

y

1 3

3

1

3

3

B A

C

x

y

2

2

4

42

2

4

4

XZ

Y

xO

y

2

2

4 622

4

4

6

M

A T

H

XZ

YB A

C

yes; preserves distance and angle measures

(a) Answers may vary. Sample: P maps to Pr; (b) PQ and P9Q9; QR and Q9R9; PR and P9R9

(a) Answers may vary. Sample: lU maps to lUr; (b) RS and R9S9; ST and S9T9; TU and T9U9; UR and U9R9

no; does not preserve distance and angle measures

324 23 1


6

Name Class Date

Th e dashed-line fi gure is a translation image of the solid-line fi gure. Write a rule to describe each translation.

8. To start, identify the coordinates of the vertices of both fi gures.

Th e vertices of the preimage are:

A(23, u), B(23, u), and C(1, u).

Th e vertices of the image are:

Ar(u, 24), Br(u, 21), and Cr(3, u).

Th e translation rule is9.

9. 10.

11. You and your friends are visiting a city with blocks laid out in a grid. You walk 7 blocks north and 3 blocks west to a restaurant. After you eat, you then walk 10 blocks east and 3 blocks south to meet up with a friend. Describe your fi nal location based on your starting point.

12. nABC has coordinates A(2, 3), B(4, 22), and C(3, 0). After a translation the coordinates of Ar are (6, 21). What are the coordinates of Br and Cr?

13. Use the graph to the right. Write three diff erent translation rules for which the image of nRST has a vertex at the origin.

4

2 44xA

B

C

A

B

C

y

x

2

4

2

6

4

6

4 6

y

xX

Y Z

W

X

Y Z

W

4

6

4

6

644 8x

A

D

B

C

y

xA

D

B

CO

4

2

6

642x

R

T

S

y

xO

9-1 Practice (continued) Form K

Translations

1 4 2

21 21 23

4 blocks north and 7 blocks east

TR5, 5S(WXYZ )

B9(8, 26), C9(7, 24)

TR23, 22S(kRST ); T

R25, 26S(kRST ); TR27, 22S(kRST )

TR2, 25S(kABC)

TR26, 24S(ABCD)


7

Name Class Date

9-1 Standardized Test Prep Translations

Multiple Choice

For Exercises 1–4, choose the correct letter.

1. In the diagram, nArBrC r is an image of nABC. Which rule describes this translation?

T,25, 23.(nABC)

T,5, 3.(nABC)

T,23, 25.(nABC)

T,3, 5.(nABC)

2. If T,3, 27.(TUVW ) 5 T rU rV rW r, what translation maps T rU rV rW r onto TUVW ?

T,3, 27.(T rU rV rW r) T,7, 23.(T rU rV rW r)

T,27, 3.(T rU rV rW r) T,23, 7.(T rU rV rW r)

3. Which of the following is true for a rigid motion?

Th e preimage and the image have the same measurements.

Th e preimage is larger than the image.

Th e preimage is smaller than the image.

Th e preimage is in the same position as the image.

4. nRSV has coordinates R(2, 1), S(3, 2), and V(2, 6). A translation maps point R to Rr at (24, 8). What are the coordinates for Sr for this translation?

(26, 24) (23, 2) (23, 9) (24, 13)

Short Response

5. nLMP has coordinates L(3, 4), M(6, 6), and P(5, 5). A translation maps point L to Lr at (7, 24). What are the coordinates for Mr and for Pr for this translation?

A

C

B

A

4

4 3

2

4

y

xB

C

[2] M9(10, 22) and P9(9, 23) [1] one correct coordinate given [0] no correct coordinates given

B

I

A

H


8

Name Class Date

9-1 Enrichment Translations

Soccer Field TranslationsBelow is a diagram of half a soccer fi eld with seven off ensive players. By passing and dribbling the ball, the players try to score by getting the ball into the goal. When a player passes, dribbles, or kicks the ball, the ball is being translated on the soccer fi eld.

Use the diagram to complete each Exercise. Each unit on the grid represents 1 unit on a coordinate grid. Assume the ball’s starting position is (x, y).

1. Garcia passes the ball to Wilson. What is the rule for this translation?

2. Young passes the ball to Foster. What is the rule for this translation?

3. Kwan passes the ball to Carmona. What is the rule for this translation?

4. Adams passes the ball to Kwan, who then passes the ball to Garcia. Write two rules to describe these translations.

5. Carmona passes the ball according to the rule T,28, 26.(x, y). Who receives Carmona’s pass?

6. Wilson kicks the ball according to the rule T,3, 26.(x, y). Is a goal scored if the goalie does not block the ball?

7. Young dribbles the ball according to the rule T,0, 210.(x, y), and then passes the ball according to the rule T,8, 24.(x, y). Who receives Young’s pass?

8. If Carmona kicks the ball toward the goal and the goalie stops it in the penalty area, which of the following rules could represent Carmona’s kick?

T,26, 212.(x, y) T,0, 212.(x, y)

T,23, 215.(x, y) T,5, 215.(x, y)

Young

Foster

Wilson

Garcia

Kwan

Carmona

Adams

GOAL LINE GOAL

CENTER LINE

GOAL AREA

PENALTY AREAPENALTY ARC

SID

ELIN

E

SID

ELIN

E

TR27, 22S(x, y)

TR4, 28S(x, y)

TR210, 7S(x, y)

Foster

yes

B

Wilson

TR4, 28S(x, y) and T

R27, 23S(x, y)


9

Name Class Date

9-1 Reteaching Translations

A translation is a type of transformation. In a translation, a geometric fi gure changes position, but does not change shape or size. Th e original fi gure is called the preimage and the fi gure following transformation is the image.

Th e diagram at the right shows a translation in the coordinate plane. Th e preimage is nABC. Th e image is nArBrCr.

Each point of nABC has moved 5 units left and 2 units up. Moving left is in the negative x direction, and moving up is in the positive y direction. So each (x, y) pair in nABC is mapped to (x 2 5, y 1 2) in nArBrCr. Th e function notation T,25, 2.(nABC) 5 nArBrCr describes this translation.

All translations are rigid motions because they preserve distance and angle measures.

Problem

What are the vertices of T,5, 21.(WXYZ)? Graph the image of WXYZ.

T,5, 21.(W) 5 (24 1 5, 1 2 1), or Wr(1, 0)

T,5, 21.(X) 5 (24 1 5, 4 2 1), or Xr(1, 3)

T,5, 21.(Y ) 5 (21 1 5, 4 2 1), or Yr(4, 3)

T,5, 21.(Z) 5 (21 1 5, 1 2 1), or Zr(4, 0)

Exercises

Use the rule to fi nd the vertices of the image.

1. T,2, 23.(nMNO) 2. T,21, 0.(JKLM )

A

CB

A

CB

2

24

2

4

y

x

X Y

W Z

X Y

W Z

2

24

y

M

O

N

3 1

2

2

y

x

J K

LM3 3

3

3y

x

M9(0, 1), N9(21, 22), O9(1, 23) J9(22, 2), K9(1, 2), L9(1, 21), M9(22, 21)


10

Name Class Date

9-1 Reteaching (continued)

Translations

Problem

What rule describes the translation that maps ABCD onto ArBrCrDr?

To get from A to Ar (or from B to Br or C to Cr or D to Dr), you move 8 units left and 7 units down. Th e translation maps (x, y) to (x 2 8, y 2 7). Th e translation rule is T,28, 27.(ABCD).

Exercises• On graph paper, draw the x- and y-axes, and label

Quadrants I–IV.

• Draw a quadrilateral in Quadrant I. Make sure that the vertices are on the intersection of grid lines.

• Trace the quadrilateral, and cut out the copy.

• Use the cutout to translate the fi gure to each of the other three quadrants.

Write the rule that describes each translation of your quadrilateral.

3. from Quadrant I to Quadrant II

4. from Quadrant I to Quadrant III

5. from Quadrant I to Quadrant IV

6. from Quadrant II to Quadrant III

7. from Quadrant III to Quadrant I

Refer to ABCD in the problem above.

8. Give the vertices of T,22, 25.(ABCD).



2

2

2

4

4

6

246 4 6

B

C

AD

A

B

C

D

y

xO

x

y

Check students’ work.




A9(0, 23), B9(1, 1), C9(4, 21), D9(5, 24)

A9(4, 22), B9(5, 2), C9(8, 0), D9(9, 23)

A9(3, 5), B9(4, 9), C9(7, 7), D9(8, 4)


Quadrant II Quadrant I

Quadrant III Quadrant IV


11

Name Class Date

9-2 Additional Vocabulary SupportRefl ections

Use the chart below to review vocabulary. Th ese vocabulary words will help you complete this page.

Related Words Explanations Examples

OrientAWR ee ent

To place or direct in a position, with reference to a certain point

Orient the box so that the top faces up.

Orientationawr ee en TAY shun

Th e way an object is positioned Th e orientation of the window is facing east.

Visual VIZH oo ul

A picture or display used to show something

A road map is a visual that can be used to fi gure out how to get to diff erent places.

Vision VIZH un

Seeing with the eyes, sight His vision is so good he can read the street sign from 60 feet away.

Visualize VIZH oo uh lyz

To form a picture of something in your mind

An artist tries to visualize a fi nished painting before starting to paint.

Refl ectrih FLEKT

To show an image of something Th e mirror refl ects light onto the wall.

Refl ectionrih FLEK shun

A mirror image of an object Th e artist can see a refl ection of the sky on the pond.

Circle the correct word to complete the sentence.

1. Th e [orient/orientation] of the solar panel is toward the sun.

2. Th e water [refl ects/refl ection] the clouds and trees around the lake.

3. Th e editor can read small print. He has good [visual/vision].

Use the vocabulary above to fi ll in the blanks.

4. He learns a lot from pictures. He is a learner.

5. She looks at her in the mirror as she combs her hair.

6. Before she plays in the game, the soccer player will herself scoring a goal.

visual

refl ection

visualize


12

Name Class Date

9-2 Think About a Plan Refl ections

Copy the pair of fi gures. Th en draw the line of refl ection you can use to map one fi gure onto the other.

1. Designate one fi gure to be the preimage and one to be its refl ection. Label two points A and B on the preimage and label their images (Ar and Br) on the refl ection.

2. Sketch AAr and BBr. Th is step is optional, but provides a frame of reference for later steps.

3. Open your compass to a size greater than half the distance between point A and Ar. With the compass at A, draw an arc above and below the approximate midpoint of AAr.

4. Do the same from point Ar. Label points P and Q at the points where the arcs intersect.

5. Draw PQ. To check your work, do the same from points B and Br.

B

B

A

A

B

B

A

A

B

B

A

A

BP

Q

B

A

A

P

Q

B

B

A

A

Answers may vary. Samples:


13

Name Class Date

9-2 Practice Form G

Refl ections

Find the coordinates of each image.

1. Rx-axis(A)

2. Ry-axis(B)

3. Ry51(C )

4. Rx521(D)

5. Ry521(E )

6. Rx52(F)

Coordinate Geometry Given points M(3, 3), N(5, 2), and O(4, 4), graph kMNO and its refl ection image as indicated.

7. Ry-axis 8. Rx-axis

9. Rx51 10. Ry522

Copy each fi gure and line <. Draw each fi gure’s refl ection image across line <.

11. 12.

13. 14.

4 2 2 4

2

2

4

4

x

y

A

C

D

EF

B

,

,

,

,

(2, 22)

(5, 23)

(24, 4)

(23, 0)

(2, 4)

(0, 2)

M

4 2 42

6

x

y

N

OM

O

N

M

4

6

2 4 x

N

O

M

ON

2

y

M

4 2

6

x

y

N

O

MO

N

N

4

6

2

x

MO

M ON

y

62

6

2


14

Name Class Date


Refl ections

Copy each pair of fi gures. Th en draw the line of refl ection you can use to map one fi gure onto the other.

15. 16.

Find the image of Z(1, 1) after two refl ections, fi rst across line <1, and then across line <2.

17. /1 i x 5 2, /2 i y-axis 18. /1 i x 5 22, /2 i x-axis

19. /1 i y 5 2, /2 i x-axis 20. /1 i y 5 23, /2 i y-axis

21. /1 i x 5 3, /2 i y 5 2 22. /1 i x 5 21, /2 i y 5 23

Use the graph at the right for Exercises 23 and 24.

23. Berit lives 3 mi east of Rt. 147 and 1 mi north of Rt. 9. Jane lives 3 mi east of Rt. 147 and 5 mi north of Rt. 9. Th e girls want to start at Berit’s house, hike to Rt. 147, then on to Jane’s house. Th ey want to hike the shortest distance possible. To which point on Rt. 147 should they walk? (Hint: First fi nd the line of refl ection if Berit’s house is refl ected onto Jane’s house.)

24. Instead of ending the hike at Jane’s house, the girls want to hike to an inn 2 mi north of Jane’s house. Th ey want to hike the shortest possible total distance, starting from Berit’s house, walking to Rt. 147, and then to the inn. To which point on Rt. 147 should they walk? (Hint: First fi nd the line of refl ection if Berit’s house is refl ected onto the inn.)

25. Point A on a coordinate grid is at (3, 4). What are the coordinates of Ry5x(A)?

26. Point Z on a coordinate grid is at (21, 3). What are the coordinates of Ry52x(Z) ?

27. Give an example of a place you may see a geometric refl ection in everyday life. Explain.

x

yNW

ES

Rt. 9Rt. 147

Jane’s house

Berit’s house

(23, 1) (25, 21)

(21, 27)

(23, 27)

(1, 23)

(5, 3)

to a point 3 mi north of Rt. 9

to a point 4 mi north of Rt. 9

Answers may vary. Samples: patterns in tiles, wallpaper, paintings

(4, 3)

(23, 1)


15

Name Class Date

9-2 Practice Form K

Refl ections

Find the coordinates of each image.

1. Rx-axis(A)

2. Ry-axis(B)

3. Ry51(C)

4. Rx521(D)

5. Ry523(E)

6. Rx522(F)

Coordinate Geometry Given points S(1, 22), T(4, 21), and V(4, 24), graph kSTV and its refl ection image as indicated.

7. Ry-axis

To start, draw the triangle and show the y-axis as the dashed line of refection.

Th en locate Sr so that the y-axis is the

perpendicular bisector of SSr.

Repeat to fi nd T r and V r.

8. Rx-axis 9. Rx521

Copy each fi gure and line <. Draw each fi gure’s refl ection image across line <.

10. 11.

224

2

4

2

4

4A

F

EBD

C

y

xO

224

4

2

4

T

V

SS

y

xO

y

xO 2 4246

4

2

2

4

4

y

xO 2 422

4

46

2

4

, ,

(4, 22)

(23, 23)

(22, 21)

(2, 23)

(1, 25)

(26, 3)

T

V

V

TS

V

S

T

V

ST

V

T

S


16

Name Class Date

Copy each pair of fi gures. Th en draw the line of refl ection you can use to map one fi gure onto the other.

12. 13.

Use the fi gure at the right to help you solve Exercises 14 and 15.

14. Shari wants to start at school, walk to Main St., then continue on to her house. She wants the whole trip to be the shortest distance possible. To which point on Main St. should she walk?

To start, draw a line of refl ection y 5 4.

Where does the line of refl ection intersect Main St.?

On the graph, draw the shortest route from school to Main St. to Shari’s house.

15. Shari decides instead to walk to the library. She wants to walk the shortest possible total distance, starting from school, walking to Main St., and then to the library. On the graph, draw a line of refl ection to help you fi nd the shortest path. Th en draw the path she should take to the library.

For Exercises 16–20, fi nd the coordinates of each image.

16. Ry5x(U )

To start, draw line /1 through U perpendicular to y 5 x. Th e slope of y 5 x is 9, so the slope of line /1 is 9.

What are the coordinates of U r so that UU r is bisected by y 5 x?

17. Ry5x(V ) 18. Ry5x(W )

19. Ry5x(X ) 20. Ry5x(Y )

y

y 4

Shari’s house (3, 6)

Library (3, 8)

Main St.

School (3, 2) Green St.

2

2

4 6

6

8

8

xO

W

N

SE

2 4

2

2

2

4

4

4

y

xU(4, 1)

V(2, 1)

W(2, 2)

X( 1, 3)

Y( 2, 5)

y x


Refl ections

at (0, 4)

U9(1, 4)

V9(1, 2)

X9(23, 21)

W9(22, 2)

Y9(25, 22)

1; 21; U9(1, 4)

U (1, 4)V (1, 2)

W ( 2, 2)

Y ( 5, 2)

X ( 3, 1)


17

Name Class Date

9-2 Standardized Test Prep Refl ections

Multiple Choice


1. In the graph at the right, what are the coordinates of Ry-axis(D)?

(3, 21) (23, 21)

(3, 1) (23, 1)

2. Th e coordinates of the vertices of nCDE are C(1, 4), D(3, 6), and E(7, 4). What are the coordinates of Ry53(D)?

(3, 26) (3, 23)

(3, 0) (3, 9)

3. What is true for an image and a preimage in a refl ection?

Th e image is larger than the preimage.

Th e image is smaller than the preimage.

Th e image and the preimage have the same orientation.

Th e image and the preimage have diff erent orientations.

4. In the graph at the right, what is the line of refl ection for nXYZ and nXrYrZr?

the x-axis

the y-axis

x 5 2

y 5 2

5. What is the image of A(3, 21) after a refl ection, fi rst across the line y 5 3, and then across the line x 5 21?

(25, 7) (25, 21)

(3, 21) (1, 25)

Extended Response

6. Th e coordinates of the vertices of parallelogram RSTU are R(0, 0), S(2, 3), T(6, 3), and U(4, 0). What are the coordinates of the vertices of Ry 5 x(RSTU )?

D2 2

2

2

x

y

Y

4 2 2 4 6

2

2

6

x

XX

ZZ

Y4

y

[4] R9(0, 0), S9(3, 2), T9(3, 6), and U9(0, 4) [3] Any three of the four coordinates are correct. [2] Any two of the four coordinates are correct. [1] Any one of the four coordinates is correct. [0] No correct coordinates are given.

C

H

H

A

D


18

Name Class Date

9-2 Enrichment Refl ections

Lines of ReflectionA line of refl ection is the line across which the preimage is refl ected to produce the image. When a refl ection is shown on a grid, you can use algebra to locate and give the equation of the line of refl ection.

Use the fi gure at the right to complete Exercises 1–6.

1. What point is halfway between J and Jr?

2. What point is halfway between K and Kr?

3. What point is halfway between L and Lr?

4. What formula did you use in Exercises 1–3?

5. Use the points found in Exercises 1–3 to draw the line of refl ection.

6. What is the equation of the line of refl ection? Use slope-intercept form.

Draw the line of refl ection on each grid. Th en give the equation of the line of refl ection.

7. 8. 9.

10. 11. 12.

4 2

2

4

6

6 y

xL

L

K

K

AJ

J6O

2

O

4

4

x

y

R

S

T

T

S

Q

R

Q

6 2

O

4

6

6

x

y

F

ED

F

D

E

642 4

2

4

6

2

4

6

xO

y

B

C

DA

C

AD

B62

4 2

2

O2

6

6

x

y

N

P

Q

M

P

M N

Q4

UU

V

V

W

W

T

T6 2

2

4

4

6

x

y

O8

2

x

y

O J

KH

G

I

H

GI

J

K

6

(23, 21)

(21, 0)

y 5 24 x 5 1

y 5 3x 1 2

y 5 2x

y 5 22x 2 2y 5 21

2 x 2 1

(5, 3)

the Midpoint Formula

y 5 12 x 1 1

2


19

Name Class Date

9-2 Reteaching Refl ections

A refl ection is a type of transformation in which a geometric fi gure is fl ipped across a line of refl ection.

In a refl ection, a preimage and an image have opposite orientations, but are the same shape and size. Because a refl ection preserves both distance and angle measure, a refl ection is a rigid motion.

Using function notation, the refl ection across line m can be written as Rm. For example, if Pr is the image of P refl ected over the line x 5 1, then Rx51(P) 5 Pr.

Problem

What are the refl ection images of nMNO across x- and y-axes? Give the coordinates of the vertices of Rx-axis(nMNO) and Ry-axis(nMNO).

Th e coordinates of the vertices of Rx-axis(nMNO) are (2, 23), (3, 27), and (5, 24). Th e coordinates of the vertices of Ry-axis(nMNO) are (22, 3), (23, 7), and (25, 4).

Exercises

Use a sheet of graph paper to complete Exercises 1–5.

1. Draw and label the x- and y-axes on a sheet of graph paper.

2. Draw a scalene triangle in one of the four quadrants. Make sure that the vertices are on the intersection of grid lines.

3. Fold the paper along the axes.

4. Cut out the triangle, and unfold the paper.

5. Label the coordinates of the vertices of the refl ection images across the x- and y-axes.

2

4

6

2 4 6x

O

y

M

N

O

fold

fold

Fold the paper along the x-axis and y-axis.

Cut out the triangle. Unfold the paper.

Copy the fi gure onto a piece of paper.







20

Name Class Date


Refl ections

To graph a refl ection image on a coordinate plane, graph the images of each vertex. Each vertex in the image must be the same distance from the line of refl ection as the corresponding vertex in the preimage.

Rx-axis describes the refl ection across the x-axis.

If P has coordinates (x, y) and Rx-axis(P) 5 Pr, then Pr has coordinates (x, 2y).

Th e x-coordinate does not change.

Th e y-coordinate tells the distance from the x-axis.

Ry-axis describes the refl ection across the y-axis.

If P has coordinates (x, y) and Ry-axis(P) 5 Pr, then Pr has coordinates (2x, y).

Th e y-coordinate does not change.

Th e x-coordinate tells the distance from the y-axis.

Problem

nABC has vertices at A(2, 4) , B(6, 4) , and C(3, 1) . What is the graph of Rx-axis(nABC)?

Step 1: Graph Ar, the image of A. It is the same distance from the x-axis as A. Th e distance from the y-axis has not changed. Th e coordinates for Ar are (2, 24).

Step 2: Graph Br. It is the same distance from the x-axis as B. Th e distance from the y-axis has not changed. Th e coordinates for Br are (6, 24).

Step 3: Graph Cr. It is the same distance from the x-axis as C. Th e coordinates for Cr are (3, 21).

Step 4: Draw nArBrCr.

Each fi gure is refl ected as indicated. Find the coordinates of the vertices for each image.

6. nFGH with vertices F(21, 3), G(25, 1), and H(23, 5) reflected by Rx-axis

7. nCDE with vertices C(2, 4) , D(5, 2) , and E(6, 3) reflected by Rx-axis

8. nJKL with vertices J(21, 25), K(22, 23), and L(24, 26) reflected by Ry-axis

9. Quadrilateral WXYZ with vertices W(23, 4), X(24, 6), Y(22, 6), and Z(21, 4) reflected by Ry-axis

B

4 62

2

2

4

4

x

yA

C

B

62

2

2

4

4

x

y

A

C

F9(21, 23) , G9(25, 21) , H9(23, 25)

C9(2, 24) , D9(5, 22) , E9(6, 23)

J9(1, 25) , K9(2, 23) , L9(4, 26)

W9(3, 4) , X9(4, 6) , Y9(2, 6) , Z9(1, 4)


21

Name Class Date

9-3 Additional Vocabulary SupportRotations

Concept List

angle of rotation center of rotation clockwise

counterclockwise rigid motion rotation

r(908, O) r(1808, O) r(2708, O)

Choose the concept from the list above that best represents the item in each box.

1. Point X 2. Th ese transformations are all examples of this.

3. Th is type of transformation

4. 908 for this transformation

5. Th is transformation maps each point (x, y) to ( y, 2x).

6.

7. 8. Th is transformation maps each point (x, y) to (2y, x).

9. Th is transformation maps each point (x, y) to (2x, 2y).

A

D

C

B

A

B D

C

Point X

A

B D

C

A

B

C

D

Point X

90

center of rotationrigid motion rotation

angle of rotation

r(2708, O)

clockwise

counterclockwise

r(908, O)

r(1808, O)


22

Name Class Date

9-3 Think About a PlanRotations

Coordinate Geometry Graph A(5, 2). Graph B, the image of A for a 908 rotation about the origin O. Graph C, the image of A for a 1808 rotation about O. Graph D, the image of A for a 2708 rotation about O. What type of quadrilateral is ABCD? Explain.

1. To fi nd the coordinates of B, you can use the rule for rotating a point 908 about the origin O in a coordinate plane. What are the coordinates of B 5 r(908, O)(A)?

2. To fi nd the coordinates of C, you can use the rule for rotating a point 1808 about the origin O in a coordinate plane. What are the coordinates of C 5 r(1808, O)(A)?

3. To fi nd the coordinates of D, you can use the rule for rotating a point 2708 about the origin O in a coordinate plane. What are the coordinates of D 5 r(2708, O)(A)?

4. Graph A, B, C, and D.

5. Draw AB, BC, CD, and DA.

6. Th e fi gure appears to be a square. How can you show that the fi gure is a square?

7. Using the Distance Formula, what are AB, BC, CD, and DA?

8. What are the slopes of AB, BC, CD, and DA?

9. What can you conclude about /A, /B, /C, and /D? Explain.

y

x

2

2

B(22, 5)

C(25, 22)

D(2, 25)

slope AB 5 slope CD 5 237; slope BC 5 slope DA 5 7

3

AB 5 BC 5 CD 5 DA 5"58

They are right angles, because AB ' BC, BC ' CD, CD ' DA, and DA ' AB.

Answers may vary. Sample: Show that

AB 5 BC 5 CD 5 DA and mlA 5 mlB 5 mlC 5 mlD 5 908.

A

B

C

D


23

Name Class Date

9-3 Practice Form G

Rotations

Copy each fi gure and point P. Draw the image of each fi gure for the given rotation about P. Use prime notation to label the vertices of the image.

1. 608 2. 908

3. 1208 4. 1808

Copy each fi gure and point P. Th en draw the image of JK for a 1808 rotation about P. Use prime notation to label the vertices of the image.

5. 6.

Point O is the center of regular hexagon BCDEFG. Find the image of the given point or segment for the given rotation.

7. r(1208, O)(F )

8. r(1808, O)(B)

9. r(3008, O)(BG)

10. r(3608, O)(CD)

11. r(608, O)(E )

12. r(2408, O)(FE )

A B

D C

A B

P

D C

A

B C

P

A

B C

P

L M

P

J

K

P

J

K

P

B C

D

EF

GO

M L

AB

C

J

KJ

K

D

D

E

CB

CD

BG

A

B

C

DA B

C


24

Name Class Date

For Exercises 13–15, kABC has vertices A(2, 2), B(3, 22), and C(21, 3).

13. Graph r(908, O)(nABC ). 14. Graph r(1808, O)(nABC ). 15. Graph r(2708, O)(nABC ).

x

y

x

y

x

y

16. Th e vertices of PQRS have coordinates P(21, 5), Q(3, 4), R(2, 24), and S(23, 22). What are the coordinates of the vertices of r(2708, O)(PQRS)?

17. Th e vertices of r(908, O)(KLMN ) have coordinates K r(23, 2), Lr(2, 3), M r(4, 22), and N r(22, 24). What are the coordinates of the vertices of KLMN ?

18. Reasoning Th e vertices of quadrilateral ABCD have coordinates A(4, 3), B(23, 4), C(24, 23), and D(3, 24). Explain how the transformation r(908, O)(ABCD) 5 BCDA can be used to show that the quadrilateral is square.

Find the angle of rotation about D that maps the solid-line fi gure to the dashed-line fi gure.

19. 20.

21. A pie is cut into 12 equal slices. What is the angle of rotation about the center that will map a piece of pie to a piece that is two slices away from it?

22. nRST has vertices at R(0, 3), S(4, 0), and T(0, 0). What are the coordinates of the vertices of r(2908, T )(nRST )?

23. nFGH has vertices F(21, 2), G(0, 0), and H(3, 21). What are the coordinates of the vertices of r(2908, G)(nFGH )?

D

D


Rotations

1138 about 3408

R9(3, 0), S9(0, 24), T 9(0, 0)

F9(22, 21), G9(0, 0), and H9(1, 3)

608

P9(5, 1), Q9(4, 23), R9(24, 22), S9(22, 3)

K(2, 3), L(3, 22), M(22, 24), N(24, 2)

Answers may vary. Sample: Because the transformation is a rigid motion, AB 5 BC 5 CD 5 DA and mlA 5 mlB 5 mlC 5 mlD. Since all angles of the quadrilateral have the same measure, the angle measure of each angle must be 908. Thus, the quadrilateral is a square by defi nition.

A

C

B

AC

B

BA

C


25

Name Class Date

9-3 Practice Form K

Rotations

Copy each fi gure and point R. Draw the image of each fi gure for the given rotation about R. Use prime notation to label the vertices of the image.

1. 308

To start, draw a 308 angle with R as the vertex

and RA as one side.

Locate Ar so that RAr >9.

Continue to fi nd Br and Cr.

2. 608 3. 908

Copy each fi gure and point R. Th en draw the image of XY for a 1208 rotation about R. Use prime notation to label the vertices of the image.

4. 5.

Point R is the center of regular quadrilateral MATH. Find the image of the given point or segment for the given rotation.

6. r(908, R)(H )

7. r(1808, R)(M )

8. r(2708, R)(AT )

9. r(3608, R)(HM )

C

B

A R

R

F

GE

D

M

A

T

H

R

RX

Y

R

X

Y

M

A

H

T

R

RA

TH

HM

T

T

B

C

A

F

DE

GM A

H

T

Y

X

X

Y


26

Name Class Date

For Exercises 10–12, ABCD has vertices A(1, 1), B(1, 3), C(4, 3), and D(4, 1).

10. Graph r(908, O)(ABCD).

To start, graph ABCD.

Ar 5 r(908, O)(A) 5 (21, u)

Br 5 r(908, O)(B) 5 (23, u)

C r 5 r(908, O)(C) 5 (u, 4)

Dr 5 r(908, O)(D) 5 (u, 4)

Th en graph Ar, Br, C r, and Dr.

11. Graph r(1808, O)(ABCD). 12. Graph r(2708, O)(ABCD).

x

y

x

y

13. Th e vertices of nPQR have coordinates P (1, 5), Q(3, 1), and R(22, 1). What are the coordinates of the vertices of r(908, O)(nPQR)?

14. ABCD has vertices A(4, 2), B(22, 2), C(24, 22), and D(2, 22). Which of the following quadrilaterals is r(1808, O)(ABCD)?

ABCD BCDA CDAB DABC

Find the angle of rotation about B that maps the solid-line fi gure to the dashed-line fi gure.

15. 16.

17. nXYZ has vertices at X(2, 0), Y(0, 0), and Z(0, 5). Find the coordinates of the vertices of r(1808, Y )(nXYZ).

x

yC

D

B

A

B

B


Rotations

23

21

C

P 9(25, 1), Q9(21, 3), R9(21, 22)

1

1

2708

X 9(22, 0), Y 9(0, 0), Z 9(0, 25)

1208

D

A

C

B

B

A

C

D

D

B

C

A


27

Name Class Date

9-3 Standardized Test Prep Rotations

Multiple Choice

In Exercises 1–5, choose the correct letter.

1. Square ABCD has vertices A(3, 3), B(23, 3), C(23, 23), and D(3, 23). Which of the following images is A?

r(908, O)(C) r(1808, O)(D) r(2708, O)(B) r(2708, O)(C )

2. r(908, O)(PQRS) has vertices Pr(1, 5),Qr(3, 22), Rr(23, 22), and Sr(25, 1). What are the coordinates of Q?

(22, 23) (23, 2) (2, 3) (23, 22)

3. Point A is the center of regular hexagon GHIJKL. What is r(3008, A)(I )?

J L

K M

4. A Ferris wheel has 16 cars spaced equal distances apart. Th e cars are numbered 1–16 clockwise. What is the measure of the angle of rotation that will map the position of car 16 onto the position of car 13?

22.58 458 67.58 908

5. Given P(2, 25), what are the coordinates of r(908, O)(P)?

(5, 2) (25, 2) (5, 22) (22, 25)

Short Response

6. nABC has coordinates A(3, 3), B(0, 0), and C(3, 0). What are the coordinates of r(1808, B)(A) and r(1808, B)(C )?

G H

I

JK

LA

C

A

H

[2] A9(23, 23) and C9(23, 0) [1] one correct coordinate given [0] no correct coordinates given

A

A


28

Name Class Date

9-3 EnrichmentRotations

Three-Dimensional RotationsRotations in three dimensions are generally covered in calculus and analytical geometry. However, a little deeper insight into rotations in two dimensions can be gained by taking a brief glimpse at rotations in three dimensions.

Th e three-dimensional coordinate system has three axes: x, y, and z, as shown at the right. Th ere are three intersecting planes, each of which is referred to by the two axes that it contains: the xy plane, the xz plane, and the yz plane. Each point in the system has coordinates in the form (x, y, z).

In two dimensions, rotations are movements around a fi xed point in a plane. A triangle rotated about the origin in two dimensions could have been “moved” just as easily by leaving the triangle as is and rotating the x- and y-axes. In fact, in three dimensions this is exactly how rotations are accomplished. Th e axes are rotated, and the geometric fi gure is mapped to a new location.

For example, the rectangular solid below has vertices at A(22, 0, 3), B(22, 2, 3), C(22, 2, 0), D(22, 0, 0), E(26, 0, 0), F(26, 0, 3), G(26, 2, 3), and H(26, 2, 0). Th e second fi gure shows the solid mapped to a new position by a clockwise 908 rotation of the x- and y-axes.

What are the coordinates of the new vertices?

1. Ar 2. Br 3. Cr 4. Dr

5. Er 6. F r 7. Gr 8. Hr

What will the coordinates be if the original rectangular solid is mapped to a new position by a clockwise 1808 rotation of the x- and y-axes?

9. Ar 10. Br 11. Cr 12. Dr

13. Er 14. F r 15. Gr 16. Hr

2

4

6

46 2 4 62

4

6

6

46

z

y

x

Three-DimensionalCoordinate System

y

x

F

z

CH

EDBG

A

x

y

z x

H C

EDB

AF

G

(2, 0, 3)

(6, 0, 0)

(0, 22, 3) (22, 22, 3) (22, 22, 0) (0, 22, 0)

(0, 26, 0) (0, 26, 3) (22, 26, 3) (22, 26, 0)

(2, 22, 3)

(6, 0, 3)

(2, 22, 0)

(6, 22, 3)

(2, 0, 0)

(6, 22, 0)


29

Name Class Date

9-3 ReteachingRotations

A turning of a geometric fi gure about a point is a rotation. Th e center of rotation is the point about which the fi gure is turned. Th e number of degrees the fi gure turns is the angle of rotation. (In this chapter, rotations are counterclockwise unless otherwise noted.)

A rotation is a rigid motion because it preserves distance and angle measure. Using function notation, the rotation of x8 with center of rotation P can be written as r(x8, P).

For example, ABCD is rotated about Z. ABCD is the preimage and ArBrCrDr is the image. Th e center of rotation is point Z. Th e angle of rotation is 828. Using function notation, r(828, Z )(ABCD) 5 ArBrC rDr.

Th e distance from the center of rotation to a point in the preimage is the same as the distance from the center of rotation to the corresponding image point. Th e measure of the angle formed by a point in the image, the center of rotation as the vertex, and the corresponding image point is equal to the angle of rotation. In the example, ZA 5 ZAr, ZB 5 ZB r, ZC 5 ZC r, ZD 5 ZDr, and m/AZAr 5 m/BZBr 5 m/CZC r 5 m/DZDr 5 828.

Exercises

Complete the following steps to draw r(808, T )(kXYZ).

1. Draw TX . Use a protractor to draw an 808 angle with

vertex T and side TX .

2. Use a compass to construct TX r > TX .

3. Locate Y ’ and Z ’ in a similar manner.

4. Draw nXrYrZr.

Copy kXYZ to complete Exercises 5–7.

5. Draw r(1208, T )(nXYZ ).

6. Draw a point S inside nXYZ. Draw r(1358, S )(nXYZ ).

7. Draw r(908, Y )(nXYZ ).

A

BC

D

A B

CD

Z82

Y

X

T

ZY

X ZX

YZ

T

Y

X Z

XY

Z

T

Y

X Z

X

Z

Y

SY

X Z

X

Z


30

Name Class Date

For a point P(x, y) in the coordinate plane, use the following rules to fi nd the coordinates of the 908, 1808, and 2708 rotations about the origin O.

r(908, O)(P) 5 (2y, x)

r(1808, O)(P) 5 (2x, 2y)

r(2708, O)(P) 5 ( y, 2x)

Problem

nFGH has vertices F (1, 1), G(2, 3), and H (4, 22). Graph nFGH and r(2708, O)(nFGH ).

First, plot F, G, and H and connect the points to graph nFGH .

Next, use the rule for a 2708 rotation to fi nd the coordinates of the vertices of r(2708, O)(nFGH ) 5 nF rGrH r.

r(2708, O)(F ) 5 (1, 21) 5 F r

r(2708, O)(G ) 5 (3, 22) 5 G r

r(2708, O)(H ) 5 (22, 24) 5 H r

Th en plot F r, Gr, and H r and connect the points to graph r(2708, O)(nFGH ).

Exercises

For Exercises 8 and 9, kABC has vertices A(2, 1), B(2, 3), C(4, 1).

8. Graph r(908, O)(nABC). 9. Graph r(1808, O)(nABC).

x

y

x

y

10. Th e vertices of nDEF have coordinates D(21, 2), E(3, 3), and F (2, 24). What are the coordinates of the vertices of r(908, O)(nDEF)?

11. Th e vertices of PQRS have coordinates P(22, 3), Q(4, 3), R(4, 23), and S(22, 23). What are the coordinates of the vertices of r(2708, O)(PQRS)?

H

G

F

H

G

F x

y


Rotations

D9(22, 21), E9(23, 3), F 9(4, 2)

P 9(3, 2), Q9(3, 24), R9(23, 24), S(23, 2)

A

C

B

AC

B


31

Name Class Date

9-4 Additional Vocabulary SupportCompositions of Isometries

A student wants to find the image of nABC for the glide reflection Rx51 + T,0, 2. .

He wrote these steps to solve the problem on note cards, but they got mixed up.

Use the note cards to write the steps in order. The completed graph is shown below right.

1. First, ______________________________________________________

2. Second, ____________________________________________________

___________________________________________________________

3. Next, _________________________________

4. Then, _________________________________

5. Then, _________________________________

______________________________________

6. Finally, ________________________________

2 4

2

2

4

4

x

y

A

B C

A

B C

A

C B

2 4

2

2

4

4

x

y

Write the rule to use for the translation. T,0, 2. maps each (x, y) to (x, y 1 2)

Find the images of A, B, and C for a reflection of the translation image.

Draw the translation image of kABC.

Draw the reflection image of kABC.

Graph the line of reflection x 5 1.

Find the translation image. The image of A is at (22, 3), the image of B is at (24, 1), and the image of C is at (21, 1).

write the rule to use for the translation. TR0, 2S maps each (x, y) to (x, y12)

find the translation image. The image of A is at (22, 3), the image of

B is at (24, 1), and the image of C is at (21, 1).

draw the translation image of kABC.

find the images of A, B, and C for a reflection

of the translation image.

draw the reflection image of kABC.

graph the line of reflection x 5 1.


32

Name Class Date

9-4 Think About a Plan Compositions of Isometries

Identify the mapping nEDC SnPQM as a translation, refl ection, rotation, or glide refl ection. Write the rule for each translation, refl ection, rotation, or glide refl ection. For glide refl ections, write the rule as a composition of a translation and refl ection.

Know

1. What strategy could you use to isolate the fi gures with which you are working?

2. What kind of mapping appears to have happened here? Explain.

Need

3. What more do we need to know to have a complete answer?

Plan

4. Write a translation rule that maps nEDC onto nPQM .

5. Write a refl ection rule that maps nEDC onto nPQM .

6. Write a rule that maps nEDC onto nPQM .

A

B

C

D

EF

G

H J

L

M

N

P

Q

x

yK

I

2

71

4

2

4

Draw a diagram of kEDC and kPQM only.

a glide refl ection; kEDC is translated from Quadrant II to Quadrant I, and then

refl ected into Quadrant IV on to kPQM

Answers may vary. Sample: Rx-axis

Answers may vary. Sample: (Rx-axis + TR11, 0S)(kEDC )

a translation rule and a refl ection rule

Answers may vary. Sample: TR11, 0S


33

Name Class Date

9-4 Practice Form G

Compositions of Isometries

Find the image of each letter after the transformation Rm + R/. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation.

1. 2.

Graph kDML and its glide refl ection image.

3. (Ry-axis + T,3, 0.)(nDML) 4. (Ry51 + T,21, 0.)(nDML)

5. (Rx52 + T,0, 1.)(nDML) 6. (Ry5x + T,3, 3.)(nDML)

7. Lines / and m intersect at point P and are perpendicular. If a point Q is refl ected across / and then across m, what transformation rule describes this composition?

8. A triangle is refl ected across line / and then across line m. If this composition of refl ections is a translation, what is true about m and /?

m

X,

mV

T ,

45

2

2 44

4

4

2

L M

D

x

y

2

2

4

4

4

2

2x

y D

L M

2

2

4

4

4

4

2x

y

D

L M

2

2

4

4

4 6

2

2x

y

D M

L

XT

M

D

L

D

ML

D

M

L

D

M L

translation; directly down; twice the distance between the two lines.

rotation; point V; 908 clockwise

r(1808, P)(Q)

< n m


34

Name Class Date

Graph AB and its image A9B9 after a refl ection fi rst across <1 and then across <2. Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

9. A(23, 4), B(21, 0); 10. A(25, 2), B(23, 6);

/1 i x 5 1; /1 i x 5 22;

/2 i y 5 21 /2 i x 5 3

11. Open-Ended Draw a quadrilateral on a coordinate grid. Describe a refl ection, translation, rotation, and glide refl ection. Th en draw the image of the quadrilateral for each transformation.

Identify each mapping as a translation, refl ection, rotation, or glide refl ection. Write the rule for each translation, refl ection, rotation, or glide refl ection. For glide refl ections, write the rule as a composition of a translation and a refl ection.

12. nABC SnDEF

13. nDEF SnGHF

14. nDEF SnIJK

15. nGHF SnIJK

P maps to P9(2, 3) by the given glide refl ection. What are the coordinates of P?

16. (Rx-axis + T,0, 2.)(P) 17. (Ry5x + T,3, 22.)(P)

B

E

G

H

F K

J

ICA

D

x

y



BB

A A

2

2

4

4

6

4 6

2

x

y

2

2

4

4

4 6

2

4

2x

y

BB

A

A

a 1808 rotation about (1, 21) a translation; 10 units to the right


TR3, 22S

r(1808, F )

Ry 5 21 + TR8, 0S

Rx 5 3

(2, 25) (0, 4)


35

Name Class Date

9-4 Practice Form K


Find the image of each fi gure after the transformation Rm + R/ . Is the resulting transformation a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

1. To start, if the lines / and m are parallel, then it is a 9.

If / and m intersect, then it is a 9.

2. 3.

Graph kABC and its glide refl ection image.

4. (Rx-axis + T,22, 0.)(nABC )

To start, translate the vertices of nABC to:

Ar(u, u), Br(u, u), Cr(u, u).

Th en, refl ect nArBrCr across u.

5. (Ry-axis + T,0, 23.)(nABC ) 6. (Ry521 + T,1, 21.)(nABC)

A

B

C D

, m

A

B

C

D

,

m 45

A B

CD

,

m

A

B

xO

y

C

2

2

2

4

4

6

24 4 6

A

x

y

BC

O

2

2

4

6

4

246 6

A

B

C

y

xO

2

2

6

4

2 246 6

translationrotation

3 5 5 1 1 1

y 5 0

translation; directly right; twice the distance between the two lines.

rotation; intersection of lines m and <; 1808

rotation; intersection of lines m and <; 908 clockwise

A

B

C D

A

BC

D

A

BC

D

A

BC

B

A

C

A

B

C


36

Name Class Date

Use the given points and lines. Graph XY and its image X9Y 9 after a refl ection fi rst across <1 and then across <2. Is the resulting transformation a translation or a rotation? For a translation, describe the distance and direction. For a rotation, tell the center of rotation and the angle of rotation.

7. X(4, 3), Y(22, 1); /1 i y 5 2; /2 i x 5 2 8. X(23, 4), Y(2, 3); /1 i y 5 2; /2 i y 5 21

9. Open-Ended Draw a quadrilateral on a coordinate grid. Draw the image of the quadrilateral for one example of each transformation.

a. refl ection b. translation c. rotation d. glide refl ection

Identify each mapping as a translation, refl ection, rotation, or glide refl ection. Write the rule for each translation, refl ection, rotation, or glide refl ection. For glide refl ections, write the rule as a composition of a translation and a refl ection.

10. trapezoid ABCD S trapezoid JICD

11. trapezoid ABCD S trapezoid NKLM

12. trapezoid CIJD S trapezoid LKNM

13. trapezoid CIJD S trapezoid TSNU

14. trapezoid KLMN S trapezoid STUN

4 6246 O

4

2

4

x

y

y 2

x 2

2 4 6246 O

4

2

4

x

y

y 2

y 1

246 4 6

4

4

x

y

A

B C I

JD U

T S

H E

G F R

ON

K L

M

P

Q



a 1808 rotation about (2, 2) a translation; 6 units down


refl ection; Rx523

translation; TR10, 0S

refl ection; Rx52

glide refl ection; Rx-axis + TR2, 0S

rotation; r(1808, (3, 0))

Y

X Y

X

Y X

YX


37

Name Class Date

9-4 Standardized Test Prep Compositions of Isometries

Multiple Choice


1. Which transformation is the same as the composition Rx 5 21 + Rx 5 2?

r(1808, (21, 2)) r(1808, (2, 21)) T,3, 0. T,6, 0.

2. What type of transformation maps nABC onto nDEF ?

translation

rotation

refl ection

glide refl ection

3. A triangle is refl ected across line / and then across line m. If the lines intersect, what type of isometry is this composition of refl ections?

translation rotation refl ection glide refl ection

4. What type of isometry is shown at the right?

translation refl ection

rotation glide refl ection

5. If (Ry 5 21 + T,0,3.)(X) 5 Xr(3, 22), what are the coordinates of X?

(25, 22) (22, 22) (22, 25) (3, 23)

Short Response

6. What type of transformation is shown? Give the translation rule, refl ection rule, rotation rule, or composition rule of the glide refl ection.

2

4

2

2x

y

D

FC B

A

E

4

4

2

4

4x

y

A

A

C

CB

B

[2] glide refl ection; Ry-axis + TR0,24S[1] Student identifi es transformation or identifi es rule. [0] incorrect or no transformation type given

D

F

B

H

D


38

Name Class Date

9-4 Enrichment Compositions of Isometries

Compositions of IsometriesYou learned that a glide refl ection is a composition of a translation and then a refl ection. A glide refl ection can also be expressed as a combination of three refl ections. Th e fi rst translation step can be accomplished by two refl ections, and the fi nal step is a third refl ection.

1. Look at the glide refl ection at the right. nABC can be mapped to nArBrCr by Ry 5 0 + T,6, 0.. It could also be mapped to nArBrCr by three refl ections. What is the glide refl ection as the composition of three refl ections?

Now consider a rotation and a refl ection. In the fi gure at the right, r(1808, O)(figure X ) 5 figure Y , and then Rx 5 22(figure Y ) 5 figure Z , so that (Rx 5 22 + r(1808, O))(figure X ) 5 figure Z. All fi gures are congruent.

2. Th is rotation and refl ection composition can be expressed in many other ways. What is another composition of a refl ection and a translation that will map X to Z?

3. Look at the rotation and refl ection composition again. Can it be expressed as a composition of refl ections? If so, what is the minimum number of refl ections that will map X to Z ?

4. Write a compostion of refl ection rules that will map X to Z.

5. Draw a glide refl ection on a coordinate grid. Write a composition of a translation and a refl ection to describe the glide refl ection. Th en express this glide refl ection as a composition of three refl ections.

6. Draw a rotation and refl ection composition on a coordinate grid. Write the composition of a rotation rule and a refl ection rule. Th en express this composition in two other ways.

2

4

2

4

24 B

A

C

AC

B x

y

2

4

2

4

24

X

Z Y

x

y

Answers may vary. Sample: Rx-axis + Rx 5 2 + Rx 5 21

Answers may vary. Sample: TR24, 0S + Rx-axis

yes; 3

Answers may vary. Sample: Rx 5 25 + Rx 5 23 + Rx-axis




39

Name Class Date

9-4 Reteaching Compositions of Isometries

Two congruent fi gures in a plane can be mapped onto one another by a single refl ection, compositions of refl ections, or glide refl ections.

Compositions of two refl ections may be either translations or rotations.

If a fi gure is refl ected across two parallel lines, it is a translation.

If a fi gure is refl ected across intersecting lines, it is a rotation.

Th e arrow is refl ected fi rst across line / and then across line m. Lines / and m are parallel. Th ese two refl ections are equivalent to translation of the arrow downward.

Th e triangle is refl ected fi rst across line / and then across line m. Lines / and m intersect at point X. Th ese two refl ections are equivalent to a rotation. Th e center of rotation is X and the angle of rotation is twice the angle of intersection, in this case, since lines / and m are perpendicular, 2 3 908, or 1808. Using function notation, (Rm + R/)(nABC) 5 r(1808, X)(nABC) 5 nArBrCr

A composition of a translation and a refl ection is a glide refl ection.

nNrOrPr is the image of nNOP, for the glide refl ection (Ry 5 21 + T,4, 21.)(nNOP) 5 nN rOrPr.

m

<

BA

C

A

CB

m

X<

Reflection

Glide

2

42

4

2N N

O

O

P

P

x

y


40

Name Class Date



Problem

What transformation maps the fi gure ABCD onto the fi gure EFGH shown at the right?

Th e transformation is a glide refl ection. It involves a translation, or glide, followed by a refl ection in a line parallel to the direction of the translation.

Exercises• Draw two pairs of parallel lines that intersect as shown at the right.

• Draw a nonregular quadrilateral in the center of the four lines.

• Use paper folding and tracing to refl ect the fi gure and its images so that there is a fi gure in each of the nine sections.

• Label the fi gures 1 through 9 as shown.

Describe a transformation that maps each of the following.

1. fi gure 4 onto fi gure 6 2. fi gure 1 onto fi gure 2




P(2, 3) maps to P9 by the given glide refl ection. What are the coordinates of P9? (Hint: it may help to graph the transformations.)

9. Rx-axis + T,3, 22. 10. Ry-axis + T,24, 2.

11. Ry5x + T,0, 23. 12. Ry54 + T,22, 23.

A

B

CD

E

F

GHA

B

CD

E

F

GH

12 3

45

6

7 89

translation

translation

refl ection

refl ection

rotation

rotation

glide refl ection

glide refl ection

(5, 21) (2, 5)

(0, 2) (0, 8)


41

Name Class Date

9-5 Additional Vocabulary SupportCongruence Transformations

Use words from the list below to complete the sentences.

angle of rotation center of rotation congruence transformation

congruent glide refl ection line of refl ection

refl ection rotation translation

1. In the coordinate plane above, all the triangles are fi gures.

2. To show that any two fi gures above are congruent, you can identify a that maps one fi gure to another.

3. A transformation that maps nABC to nDEF is a that slides nABC four units to the right and two units down.

4. A transformation that maps nDEF to nGHI is a with of 180° and at the origin.

5. A transformation that maps nGHI to nJKL is a with of x 5 4.

6. A transformation that maps nABC to nJKL is a by sliding nABC twelve units to the right and two units down and then refl ecting across the x-axis.

x

FD

A

B

C

E I

H

GJ

L

K

y

O

congruent

congruence transformation

translation

rotationcenter of rotationangle of rotation

refl ection

glide refl ection

line of refl ection


42

Name Class Date

9-5 Think About a Plan Congruence Transformations

Use congruence transformations to prove the Isosceles Triangle Th eorem.

Given: FG > FHProve: /G > /H

1. Since you want to show that /G > /H , you want to fi nd a congruence transformation that will map /G to /H . If the preimage of this transformation is nFGH , what is the image?

2. Identify the congruence transformation as a translation, refl ection, rotation, or a composition of rigid motions.

3. For each rigid motion used in the transformation, describe the translation rule, the line of refl ection, or the center and angle of rotation.

4. Explain why this transformation maps nFGH to nFHG.

5. Since this transformation maps point F to point F, point G to point H, and point H to point G, you know that it maps /G to /H . Why does it show that /G > /H ?

F

H

G

The line of refl ection is the perpendicular bisector of GH.

Let M be the midpoint of GH. Since the line of refl ection is the perpendicular bisector,

GM O HM and the refl ection maps G to H and H to G, so it maps GH to HG. Because

FG O FH and distance is preserved under refl ections, the refl ection maps FG to FH .

The transformation maps kFHG to kFGH. Because a refl ection is a rigid motion, it

means both distance and angle measure are preserved. lG and lH are congruent

because corresponding parts of congruent fi gures are congruent.

refl ection

kFHG


43

Name Class Date

9-5 Practice Form G

Congruence Transformations

For each coordinate grid, identify a pair of congruent fi gures. Th en determine a congruence transformation that maps the preimage to the congruent image.

1. 2.

3. 4.

Find a congruence transformation that maps kABC tokDEF .

5. 6.

x

B E

D

G

H

AC F

I

y

Ox

IJ

K E

FA

B C

DG

HL

y

O

x

E

F

A

BD

C y

O x

A

B

CF E

D

IG

H

y

xA

B

C F

E

D

y

Ox

AB

C

F

E

D

y

O

kABC O kGHI ; Answers may vary. Sample: r(1808, O)(kABC) 5 kGHI

CD O EF ; Answers may vary. Sample: Ry51(CD) 5 EF

Answers may vary. Sample: (TR1, 0S + Ry-axis + r(908, O))(kABC) 5 kDEF

kABC O kIHG; Answers may vary. Sample: (Ry5 2 + TR6, 0S)(kABC) 5 kIHG

Answers may vary. Sample: (TR2, 0S + r(1808, O))(kABC) 5 kDEF

ABCD O LIJK ; Answers may vary. Sample: T

R25, 24S(ABCD) 5 LIJK


44

Name Class Date

7. Verify the ASA Postulate for triangle congruence by using congruence transformations.

Given: /A > /K, /B > /L,AB > KL Prove: nABC > nKLM

8. Verify the SSS Postulate for triangle congruence by using congruence transformations.

Given: XY > RS, YZ > ST, ZX > TR Prove: nXYZ > nRST

Determine whether the fi gures are congruent. If so, describe a congruence transformation that maps one to the other. If not, explain.

9. 10.

11. Error Analysis For the fi gure at the right, a student says that nABC > nDEF because Ry-axis(nABC) 5 nDEF . What is the student’s error?

12. Reasoning Suppose one congruence transformation maps nLMNonto nPQR and another congruence transformation maps nLMN onto nPRQ. How would you classify the two triangles? Explain.

A

C

B

L

M

K

X

YZ

T

SR

x

A C D F

B Ey

O

9-5 Practice Form G


Answers may vary. Sample: Translate kABC up until points B and L coincide. Then refl ect across the vertical line containing B and L.

Answers may vary. Sample: Translate kXYZ so that points Z and T coincide. Then rotate kXYZ about point Z until YZ and ST coincide.

Not congruent; one fi gure is smaller than the other fi gure.

The transformation Ry-axis maps kABC onto kFED not kDEF , but T

R4, 0S(kABC) 5 kDEF .

Both kLMN and kPQR are isosceles triangles. Since congruence transformations exist, kLMN O kPQR and kLMN O kPRQ. By the transitive property, kPQR O kPRQ. Then PQ O PR since they are corresponding parts. By defi nition, kPQR is isosceles and so is kLMN.

Congruent; answers may vary. Sample: draw a line between the two fi gures and refl ect across the line.


45

Name Class Date

9-5 Practice Form K


For each coordinate grid, identify a pair of congruent fi gures. Th en determine a congruence transformation that maps the preimage to the congruent image.

1. Compare each pair of fi gures on the grid.

Is nABC > nDEF?

Is nDEF > nGHI ?

Is nABC > nGHI?

Now, determine a congruence transformation that maps one of the congruent triangles to the other.

2. 3.

Use congruence transformations to verify that kABC O kDEF.

4. 5.

x

BD

E

FCG

I

H

Ay

O

x

A

CR

B

M

N

PST

U

LD

y

OxM

G

H

P

QL

y

x

C

A

B

F E

D

y

Ox

FD

E

A B

C

y

O

ABCD O LMNP ; Answers may vary. Sample: (T

R4, 4S + r(908, O)(ABCD) 5 LMNP

Answers may vary. Sample: (TR0,22S + Ry2axis)(kABC)5 kDEF

LM O QP ; Answers may vary. Sample: (Ry2axis + TR1, 0S(LM) 5 QP

Answers may vary. Sample: (TR1,24S + r(908, O)(kABC)5 kDEF

Answers may vary. Sample: TR22,24S(kDEF) 5 kGHI

no

yes

no


46

Name Class Date

9-5 Practice Form K


6. Verify the SAS Postulate for triangle congruence by using congruence transformations.

Given: /G > /M, FG > LM, GH > MNProve: nFGH > nLMN


Given: /P > /S, /Q > /T, PQ > STProve: nPQR > nSTU

Determine whether the fi gures are congruent. If so, describe a congruence transformation that maps one to the other. If not, explain.

8. 9.

10. To show that nABC on the right is an equilateral triangle, what congruence transformation can you use that maps the triangle to itself? Explain.

F

H

G M

N

L

P

Q

R

US

T

A

BC

Answers may vary. Sample: Refl ect kFGH across the perpendicular bisector of GM so lG and lM coincide along with congruent side.

Answers may vary. Sample: Translate kPQR so that points R and U coincide. Then rotate kPQR about point U until PQ and ST coincide.

Answers may vary. Sample: r(608, P)(kABC) 5 BCA , where P is a point of concurrency of the triangle. As a result of that transformation and the transitive property, AB O BC O CA and lA O lB O lC .

Congruent; answers may vary. Sample: draw a line between the two fi gures and refl ect across the line.

Not congruent; one fi gure is smaller than the other fi gure.


47

Name Class Date

9-5 Standardized Test Prep Congruence Transformations

Multiple Choice


1. Which congruence transformation maps nABC to nDEF ?

T,5, 25. Rx-axis + T,5, 0.

r(1808, O) Ry-axis + r(908, O)

2. Which congruence transformation does not map nABC to nDEF ?

r(1808, O) Rx-axis + Ry-axis

T,0, 6. + Ry-axis Ry-axis + Rx-axis

3. Which of the following best describes a congruence transformation that maps nFGH to nLMN ?

a refl ection only

a translation only

a translation followed by a refl ection

a translation followed by a rotation

Short Response

4. What is a congruence transformation that maps ABCD to RSTU?

xB

EF

D

C

A y

O

x

F

E

D

CA

By

O

G

F H

N

M L

x

AB

C

DS T

UR

y

O

C

G

D

[2] TR0,25S + r(908, O))(ABCD) or equivalent congruence

transformation [1] gives one correct transformation of a possible composition of transformations that maps ABCD to RSTU [0] incorrect or no response


48

Name Class Date

9-5 Enrichment Congruence Transformations

Inverse TransformationsWhen two fi gures are congruent, there is a congruence transformation that maps the fi rst fi gure to the second fi gure and another transformation that maps from the second to the fi rst fi gure. An inverse transformation of a given congruence transformation is a transformation which maps back to the fi rst fi gure.

You can use the following rules for translations, refl ections, rotations, and compositions to fi nd an inverse transformation given a congruence transformation. For a translation T,h, k. , an inverse transformation is T,2h, 2k. . For a refl ection R/, an inverse transformation is the refl ection itself R/. For a rotation r(x8, P), an inverse transformation is r(2x8, P) or r(3608 2 x8, P). For compositions, fi nd the inverse transformation for each transformation of the composition and then compose in reverse order.

Look at the fi gure at the right. T,25, 21. maps nABC to nDEF , so T,5, 1. maps nDEF back to nABC . Rx-axis maps nDEF to nGHI , so Rx-axis maps nGHI back to nDEF . r(908, O) maps nGHI to nJKL, so r(2908, O) or r(2708, O) maps nJKL back to nGHI . Th e composition r(908, O) + Rx-axis maps nDEF to nJKL, so Rx-axis + r(2708, O) maps nJKL back to nDEF .

For Exercises 1–6, write an inverse transformation that maps kSTU to kPQR given the congruence transformation that maps kPQR to kSTU .

1. T,4,22.(nPQR) 5 nSTU 2. Ry51(nPQR) 5 nSTU

3. r(1208, O)(nPQR) 5 nSTU 4. Ry-axis + r(2108, O))(nPQR) 5 nSTU

5. (T,23, 1. + Rx-axis)(nPQR) 5 nSTU 6. (r(458, O) + T,24, 21.)(nPQR) 5 nSTU

For Exercises 7–9, use the fi gure at the right.

7. What is a glide refl ection that maps nABC to nDEF ?

8. Using the rules described above, what is an inverse transformation of the glide refl ection?

9. If you reverse the order of composition of the transformation in Exercise 8, you get a glide refl ection. Does that map nDEF to nABC?

x

AD

GH

IL

K

J

F

E

C

By

O

xA

B

C

D

E

F

y

O

TR24, 2S(kSTU) 5 kPQR

r(2408, O)(kSTU) 5 kPQR

(Rx-axis + TR3, 21S)(kSTU) 5 kPQR (TR4, 1S + r(3158, O))(kSTU) 5 kPQR

Rx-axis + TR5, 0S

TR25, 0S + Rx-axis

yes

(r(1508, O) + Ry-axis)(kSTU) 5 kPQR

Ry51(kSTU) 5 kPQR


49

Name Class Date

9-5 Reteaching Congruence Transformations

Because rigid motions preserve distance and angle measure, the image of a rigid motion or composition of rigid motions is congruent to the preimage. Congruence can be defi ned by rigid motions as follows.

Two fi gures are congruent if and only if there is a sequence of one or more rigid motions that map one fi gure onto the other.

Because rigid motions map fi gures to congruent fi gures, rigid motions and compositions of rigid motions are also called congruence transformations. If two fi gures are congruent, you can fi nd a congruence transformation that maps one fi gure to the other.

ProblemIn the fi gure at the right, nPQR > nSTU . What is a congruence transformation that maps nPQR to nSTU ?

nSTU appears to have the same shape and orientation as nPQR, but rotated 90°, so start by applying the rotation r(908, O) on the vertices of nPQR.

r(908, O)(P) 5 (24, 1), r(908, O)(Q) 5 (21, 4), r(908, O)(r) 5 (22, 1)

Graph the image r(908, O)(nPQR). A translation of 1 unit to the right and 5 units down maps the image to nSTU .

Th erefore, (T,1, 25. + r(908, O))(nPQR) 5 nSTU .

Exercises

Find a congruence transformation that maps kABCto kDEF .

1. 2.

xT

R

P

Q

US

y

O

xT

R

P

Q

US

y

O

x

AB

DE

F

C

y

Ox

A B

C

F

D

E

y

O

Answers may vary. Sample: (Rx-axis + TR5, 0S)(kABC)5 kDEF

Answers may vary. Sample: (r(2708, O) + Ry-axis)(kABC)5 kDEF


50

Name Class Date

9-5 Reteaching Congruence Transformations

If you can show that a congruence transformation exists from one fi gure to another, then you have shown that the fi gures are congruent.

Problem

Verify the SSS Postulate by using a congruence transformation.

Given: JK > RS, KL > ST, LJ > TRProve: nJKL > nRST

Start by translating nJKL so that points J and R coincide.

Because you are given that JK > RS, there is a rigid motion that maps

JK onto RS by rotating nJKL about point R so that JK and RS coincide. Th us, there is a congruence transformation that maps nJKL to nRST , so nJKL > nRST .

Exercises

3. Verify the SAS Postulate for triangle congruence by using congruence transformations.

Given: /R > /X, RS > XY, ST > YZ Prove: nRST > nXYZ


Given: /A > /J, /B > /K, AB > JKProve: nABC > nJKL

S

R TJ

K

L

J K

L

S

R T

J

K

L

S

RT

X

Z

Y

RT

S

B

A

CJ

KL

Answers may vary. Sample: Since RS O XY , translate kRST so RS coincides with XY . Then refl ect kRST across XY to complete the transformation.

Answers may vary. Sample: Translate kABC so that points C and L coincide. Then rotate kABC about point L until AB and JK coincide.


51

Name Class Date

9-6 Additional Vocabulary SupportDilations

Th e column on the left shows the steps used to graph a dilation. Use the column on the left to answer each question in the column on the right.

Problem Graphing a Dilation

What are the images of the vertices of kABC for a dilation centered at the origin with a scale factor of n 5 2? Graph the image of kABC.

1. Read the example. What do you need to fi nd to solve the problem?

Identify the coordinates of each vertex of nABC.

A(0, 3), B(22, 0), C(2, 22)

2. Th e dilation center is the origin. What are the coordinates of the origin?

Use the dilation rule where n is the scale factor.

Dn(x, y) 5 (nx, ny)

D2(x, y) 5 (2x, 2y)

3. What is a dilation rule for a transformation?

Find the images of A, B, and C.

D2(A) 5 (2 ? 0, 2 ? 3), or Ar(0, 6)

D2(B) 5 (2 ? (22), 2 ? 0), or Br(24, 0)

D2(A) 5 (2 ? 2, 2 ? (22)), or Cr(4, 24)

4. How was the rule used to fi nd the images of each vertex?

Graph Ar, Br, and Cr. Then draw nArBrCr. 5. What does it mean to graph a point?

6. Use the grid to the left to graph Ar, Br, and Cr and draw ArBrCr.

2 2

2

x

yA

B

C

24 2 4

2

4

2

4

6

xO

y

the coordinates of the vertices for

the dilation

a rule that describes how to map a

preimage onto its image

The coordinates of the vertex were

substituted into the rule to fi nd the

coordinates of the image.

(0, 0)

A

B

C

to plot the point on a coordinate grid


52

Name Class Date

9-6 Think About a Plan Dilations

Reasoning You are given AB and its dilation image ArBr with A, B, Ar, and Br noncollinear. Explain how to fi nd the center of dilation and scale factor.

Know

1. What do you know about the relationship of A, Ar, and C, the center of dilation?

2. Is this true also of B, Br, and C?

3. What relationship exists between the lengths of the segments ArBr and AB?

Need

4. Is it possible to answer this question using a specifi c point C and a specifi c scale factor? Explain.

Plan

5. How do you fi nd the center of dilation?

6. How do you fi nd the scale factor?

yes

They are collinear.

Their ratio is the scale factor.

No; there is not enough information given. The answer can only be given in

general terms.

Draw a line connecting A and A9 and a second line connecting B and B9. Point C will be

at the point where the two intersect.

Simplify the ratio of length of image

length of preimage, or A9B9AB .


53

Name Class Date

9-6 Practice Form G

Dilations

Th e solid-line fi gure is a dilation of the dashed-line fi gure. Th e labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Th en fi nd the scale factor of the dilation.

1. 2.

3. 4.

5. 6.

7. 8.

You look at each object described in Exercises 9–11 under a magnifying glass. Find the actual dimension of each object.

9. Th e image of a ribbon is 10 times the ribbon’s actual size and has a width of 1 cm.

10. Th e image of a caterpillar is three times the caterpillar’s actual size and has a width of 4 in.

11. Th e image of a beetle is fi ve times the beetle’s actual size and has a length of 1.75 cm.

12. nP rQrRr is a dilation image of nPQR. Th e scale factor for the dilation is 0.12. Is the dilation an enlargement or a reduction?

X 3 2

4

4

W

16

9

Z

B3.5

14

P

T

x

y

C

15

5X

enlargement; 53 reduction; 12

enlargement; 169

enlargement; 4

enlargement; 2

enlargement; 3

enlargement; 2

0.1 cm

43 in.

0.35 cm

reduction; 13

reduction


54

Name Class Date

A dilation has center (0, 0). Find the image of each point for the given scale factor.

13. X(3, 4); D7(X) 14. P(23, 5); D1.2(P)

15. Q(0, 4); D3.4(Q) 16. T(22, 21); D4(T)

17. S(5, 26); D53(S) 18. M(2, 2); D5(M)

19. A square has 16-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor 0.8.

20. Graph pentagon ABCDE and its image ArBrCrDrEr for a dilation with center (0, 0) and a scale factor of 1.5. Th e vertices of ABCDE are: A(0, 3), B(3, 3), C(3, 0), D(0, 23), E(21, 0).

Copy kBCD and point X for each of Exercises 21–23. Draw the dilation image kB9C9D9.

21. D(1.5, X)(nBCD)

22. D(1.5, B)(nBCD) 23. D(0.8, C)(nBCD)

B

X

C

D


Dilations

(21, 28)

(0, 13.6)

(253 , 210)

(23.6, 6)

(28, 24)

(10, 10)

The dilation image will be a square with 12.8-cm sides that shares the vertex that is the dilation center with the original square. The sides will be parallel to or along the original sides.

B

B

X

C C D

DD

B

X

C

C D

y

E

A

D

B

A

D

E

B

Cx

C

BB

X

C

CD

D


55

Name Class Date

9-6 Practice Form K

Dilations

Th e dashed-line fi gure is a dilation image of the solid-line fi gure. Th e labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Th en fi nd the scale factor of the dilation.

1. To start, identify whether the image is larger or smaller than the preimage.

Next, fi nd the scale factor:

image lengthpreimage length 5

uu

2. 3.

4. 5.

A dilation has center (0, 0). Find the image of each point for the given scale factor.

6. B(2, 3); D6(B) To start, multiply each coordinate by the scale factor.

D6(B) 5 (2 ?u, 3 ?u) 5 (u, u)

7. C(7, 2); D3(C) 8. U(23, 22); D3(U)

9. A(0, 25); D25(A) 10. G(3, 1); D24(G)

11. R(23, 5); D14(R) 12. XA12, 23B; D22(X)

D

1 in.

2 in.

8

4

D

3

6

D

9

6

D

4 4D

1

2reduction

enlargement; 2

reduction; 23

(12, 18) 6 6 12 18

(29, 26)(21, 6)

(0, 22) (212, 24)

(21, 6)(234, 11

4)

enlargement; 3

enlargement; 2


56

Name Class Date


Dilations

You look at each object described in Exercises 13–15 under a magnifying glass. Find the actual dimension of each object.

13. Th e image of a bug is 5 times its actual size and has a width of 1.5 cm. image length 5 scale factor ? actual length

9 59 ?9

14. Th e image of a worm is 4 times its actual size and has a length of 7 cm.

15. Th e image of a hair is 10 times its actual size and has a length of 0.4 cm.

16. A dilation maps nQRS to nQrRrSr. QR 5 10 in. and QrRr 5 12 in. If RS 5 12 in., what is RrSr?

17. A dilation on a coordinate grid has center (0, 0) and scale factor 2.5. Point A is at (3, 7). What is the y-coordinate of the image of A?

18. nArBrCr is a dilation image of nABC . Th e scale factor for the dilation is 1.25. Is the dilation an enlargement or a reduction?

19. A square has 12-cm sides. Describe its image for a dilation with center at one of the vertices and scale factor 0.4.

20. Graph pentagon PENTA and its image D0.5(PENTA) 5 PrErNrT rAr. Th e vertices of PENTA are: P(0, 4), E(6, 6), N(4, 0), T(0, 24), A(22, 0).

A dilation maps kMNO onto kM9N9O9. Find the missing values.

21. MN 5 2 in., MrNr 5 3.5 in. 22. MN 5 2 cm, MrNr 5 1.6 cm

NO 5 3 in., NrOr 5 z z in. NO 5 5 cm, NrOr 5 z z cm

MO 5 4 in., MrOr 5 z z in. MO 5 6 cm, MrOr 5 z z cm

2

2

4

6

2

2

4

4 4 6

y

xOA

T

T

N

E

N

EP

P

A

1.5 cm; 5; x; 0.3 cm

74 cm

0.04 cm

14.4 in.

17.5

5.25 4.0

7.0 4.8

enlargement

The dilation image will be a square with 4.8-cm sides that shares the vertex that is the dilation center with the original square. The sides will be parallel to or along the original sides.


57

Name Class Date

9-6 Standardized Test Prep Dilations

Gridded Response

Th e solid-line fi gure is a dilation of the dashed-line fi gure. Th e labeled point is the center of dilation. Find the scale factor for each dilation. Use whole numbers or decimals. Enter your responses on the grid provided.

1. 2.

Solve the problem and enter your response on the grid provided.

3. Th e image of an eraser in a magnifying glass is three times the eraser’s actual size and has a width of 14.4 cm. What is the actual width in cm?

4. A square on a transparency is 1.7 in. long. Th e square’s image on the screen is 11.05 in. long. What is the scale factor of the dilation?

5. A dilation maps nLMN to nLrMrNr. MN 5 14 in. and MrNr 5 9.8 in. If LN 5 13 in., what is LrNr?

3 6X

Z

15

20

Answers

1. 2. 3. 4. 5.

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

3 0 . 7 5 4 . 8 9 . 16 . 5


58

Name Class Date

9-6 Enrichment Dilations

Three-Dimensional Dilations

Refer to the cube in Figure 1 and its dilation in Figure 2 to complete the following exercises.

1. Give the coordinates of all vertices of the preimage. Remember, each point in the three-dimensional coordinate system has coordinates in the form (x, y, z).

a. A b. B c. C d. D e. E f. F g. G h. H

2. Give the coordinates of all vertices of the image. a. Ar b. Br c. Cr d. Dr e. Er f. F r g. Gr h. Hr

3. What is the center of the dilation?

4. What is the scale factor?

5. How do the lengths of the edges of the cubes compare?

6. How do the faces of the cubes compare?

7. How do the total surface areas of the cubes compare?

8. How do the volumes of the cubes compare?

9. How do the dilations in three dimensions appear to compare with dilations in two dimensions?

D

E H

F GA

B Cy

z

x

Figure 1

D

E H

F G

A

B C

y

z

x

Figure 2

(2, 0, 2)(0, 2, 2)(2, 0, 0)(0, 2, 0)

(4, 0, 4)

(0, 0, 0)

2

Each edge in the image is double that in the preimage.

Surface area of image 5 96 sq. units; surface area of preimage 5 24 sq. units; the ratio of the surface areas is 4 i 1.

Volume of image 5 64 cubic units; volume of preimage 5 8 cubic units; the ratio of the volumes is 8 i 1.

Dilations in three dimensions also scale each dimension by a common scale factor, like dilations in two dimensions.

Samples: Three pairs of faces are coplanar for image and preimage; three pairs of faces are parallel; each face in the image has two times the perimeter and four times the area of the corresponding face in the preimage.

(0, 4, 4)(4, 0, 0)(0, 4, 0)

(0, 0, 2)(2, 2, 2)(0, 0, 0)(2, 2, 0)

(0, 0, 4)(4, 4, 4)(0, 0, 0)(4, 4, 0)


59

Name Class Date

9-6 Reteaching Dilations

A dilation is a transformation in which a fi gure changes size. Th e preimage and image of a dilation are similar. Th e scale factor of the dilation is the same as the scale factor of these similar fi gures.

To fi nd the scale factor, use the ratio of lengths of corresponding sides. If the scale factor of a dilation is greater than 1, the dilation is an enlargement. If it is less than 1, the dilation is a reduction.

Using function notation, a dilation with center C and scale factor n . 0 can be written as D(n, C). If the dilation is in the coordinate plane with center at the origin, the dilation with scale factor n can be written as Dn. For a point P(x, y), the image is Dn(P) 5 (nx, ny).

Problem

nXrYrZr is the dilation image of nXYZ. Th e center of dilation is X. Th e image of the center is itself, so X r 5 X .

Th e scale factor, n, is the ratio of the lengths of corresponding sides.

n 5 XrZrXZ 5

3012 5

52

Th is dilation is an enlargement with a scale factor of 52.

Exercises

For each of the dilations below, A is the center of dilation. Tell whether the dilation is a reduction or an enlargement. Th en fi nd the scale factor of the dilation.

1. 2.

3. AB 5 2; ArBr 5 3 4. DE 5 3; DrEr 5 6

X X

12

18

Z

Y Y

Z

10

BB

CC

8

A A

10

A A

BB

CC

8

A A

B

B

D

C

D

C

B

D

A

B

CE

D

CE

enlargement; 54enlargement; 32 enlargement; 2

reduction; 45


60

Name Class Date


Dilations

Problem

Quadrilateral ABCD has vertices A(22, 0), B(0, 2), C(2, 0), and D(0, 22).Find the coordinates of the vertices of D2(ABCD). Th en graph ABCD and its imageD2(ABCD).

To fi nd the image of the vertices of ABCD, multiply the x-coordinates and y-coordinates by 2.

D2(A) 5 (2 ? (22), 2 ? 0) 5 Ar(24, 0) D2(B) 5 (2 ? 0, 2 ? 2) 5 Br(0, 4)

D2(C) 5 (2 ? 2, 2 ? 0) 5 Cr(4, 0) D2(D) 5 (2 ? 0, 2 ? (22)) 5 Dr(0, 24)

Exercises

Use graph paper to complete Exercise 5.

5. a. Draw a quadrilateral in the coordinate plane.

b. Draw the image of the quadrilateral for dilations centered at the origin with scale factors 12, 2, and 4.

Graph the image of each fi gure for a dilation centered at the origin with the given scale factor.

6. 7.

8. 9.

4 2 2

2

4

x

y

OA A

B

C

D

B

C

D

4 4

4

4

x

y

O

scale factor 2 scale factor 12

4 2 2 4

2

2

4

4

x

y

O

4 2 2 4

4

4

x

y

O

scale factor 23 scale factor 3

2

4 2 2 4

4

4

2

x

y

O


Check students’ graphs.


61

Name Class Date

9-7 Additional Vocabulary SupportSimilarity Transformations

Is there a similarity transformation that maps ΔABC to ΔRST? If so, identify the similarity transformation. Use words from the list below to complete the sentences and answer the question.

composition dilation line of refl ection

refl ection scale factor similar

similarity transformation translation

1. nABC and nRST appear to be fi gures.

2. Two fi gures are similar if and only if there is a that maps one fi gure to another.

3. A similarity transformation is a of rigid motions and dilations.

4. Since nRST is an enlargement of nABC , perform a with center at the origin and a of 2.

5. Since the resulting image and nRST have opposite orientations, perform a across the x-axis, which is the .

6. Finally, apply a that slides the second image seven units to the left to nRST .

x

R T

ABC

S

y

O

x

R T

A

A

B

CBC

S

y

O

x

R T

A

A

B

C

BC

S

y

O

similar

similarity transformation

composition

dilationscale factor

refl ectionline of refl ection

translation


62

Name Class Date

9-7 Think About a Plan Similarity Transformations

A printing company enlarges a banner for a graduation party by a scale factor of 8.

a. What are the dimensions of the larger banner?

b. How can the printing company be sure that the enlarged banner is similar to the original? Explain.

1. What is the scale factor for this problem?

2. What can you do with the scale factor to fi nd the dimensions of the large banner?

3. What is the width of the larger banner?

4. What is the height of the larger banner?

5. What kind of transformation did the printing company apply to the original banner in order to produce the larger banner?

6. Explain how the printing company can be sure that the enlarged banner is similar to the original.

13 in.

3 in.

Answers may vary. Sample: Dilation is a similarity transformation

so the image (the larger banner) should be similar to the original banner.

Multiply the dimensions of the original banner by the scale factor.

104 in.

24 in.

dilation

8


63

Name Class Date

9-7 Practice Form G Similarity Transformations

kABC has vertices A(22, 2), B(2, 0), and C(1, 22). For each similarity transformation, draw the image.

1. D2 + Ry-axis 2. r908 + D1.5

3. D0.5 + T,2, 22. 4. r2708 + D2 + Rx-axis

For each graph, describe the composition of transformations that maps kDEF to kLMN .

5. 6.

x

y

Ox

y

O

x

y

Ox

y

O

M

NL

F

E

D x

y

O

F

E

D

L

NM

x

y

O

Answers may vary. Sample: (Rx-axis + D2)(kDEF)5 kLMN

Answers may vary. Sample: (D0.5 + r(1808, O))(kDEF)5 kLMN

C

BA

C

B

A

C

B

A

C

B

A


64

Name Class Date

For each pair of fi gures, determine if there is a similarity transformation that maps one fi gure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain.

7. 8.

Determine whether or not each pair of fi gures below are similar. Explain your reasoning.

9. 10.

11. Reasoning A transformation maps each point (x, y) of nABC to the point (24x 1 3, 24y 1 2). Is nABC similar to the image of nABC? Explain.

12. Reasoning Do the compositions T,4, 22. + D2 and D2 + T,2, 21. describe the same similarity transformation? Explain.

10

7

3

4

A BP Q

S RD C 45

30 60

40

J

L K

H

G I

9-7 Practice Form GSimilarity Transformations

Not similar; the ratio of the corresponding longer sides is not the same as the ratio of the corresponding shorter sides.

Similar; answers may vary. Sample: refl ect the smaller fi gure across a vertical line, translate the small fi gure so that their tips meet and then enlarge by some scale factor until the fi gures coincide.

Yes; the transformation is equivalent to the similarity transformation TR3, 2S + r(1808, O) + D4, since (T

R3, 2S + r(1808, O) + D4)(x, y) 5 (TR3, 2S + r(1808, O))(4x, 4y)

5 (TR3, 2S)(24x, 24y) 5 (24x 1 3, 24y 1 2)

Yes; for each point (x, y), (TR4, 22S + D2)(x, y) 5 T

R4,22S(2x, 2y) 5 (2x 1 4, 2y 2 2) and (D2 + T

R2, 21S)(x, y) 5 D2(x 1 2, y 2 1) 5 (2(x 1 2), 2(y 2 1)) 5 (2x 1 4, 2y 2 2), so both transformations map each point (x, y) to (2x 1 4, 2y 2 2).

Similar; answers may vary. Sample: translate kGHI so that vertices I and L coincide. Then rotate kGHI 2708 about the point L, and dilate with a scale factor of 43 and center at L; kGHI,kJKL

Not similar; the sides are not in proportion.


65

Name Class Date

9-7 Practice Form K

Similarity Transformations

kLMN has vertices L(0, 2), M(2, 2), and N(0, 1). For each similarity transformation, draw the image.

1. D2 + Rx-axis

Find the image of each vertex.

(D2 + Rx-axis)(L) 5 D2(0, 22) 5

(D2 + Rx-axis)(M) 5 D2(2, 22) 5

(D2 + Rx-axis)(N) 5 D2(0, 21) 5

Th en plot the points and draw the image.

2. D0.5 + r(908, o) 3. T ,23, 22. + D1.5

For each graph, describe the composition of transformations that maps kABCto kFGH .

4. 5.

x

y

O

x

y

x

y

O

C

BA

G

H

Fy

OC

BA

FH

G x

y

O

Answers may vary. Sample: (D1.5 + Ry-axis)(kABC)5 kFGH

Answers may vary. Sample: (r(908, O) + D0.5)(kABC)5 kFGH

(0, 24)

(4, 24)

(0, 22)

N

ML

L M

N

M

NL


66

Name Class Date

9-7 Practice Form K

Similarity Transformations

For each pair of fi gures, determine if there is a similarity transformation that maps one fi gure onto the other. If so, identify the similarity transformation and write a similarity statement. If not, explain.

6. 7.

Determine whether or not each pair of fi gures below are similar. Explain your reasoning.

8. 9.

10. Reasoning A similarity transformation is described by the composition T,2, 1. + D3 . If the order of the composition is changed to be D3 + T,2, 1. , does that describe the same transformation? Explain.

B

C

A

D

P

Q

N

M

6 6

9 4

K

J

L

N

M

Similar; answers may vary. Sample: rotate the larger rectangle by 908, dilate with a scale factor of 23, and then translate so vertices A and M coincide; ABCD | MNPQ

Similar; answers may vary. Sample: rotate kLMN 1808 so that lJLK and lMLN coincide. Then dilate with some scale factor of k and center at L so that MN and JK coincide; kLMN | kLJK

No; for each point (x, y), (TR2, 1S + D3)(x, y) 5 T

R2, 1S(3x, 3y) 5 (3x 1 2, 3y 1 1), while (D3 + T

R2, 1S)(x, y) 5 D3(x 1 2, y 1 1) 5 (3(x 1 2), 3(y 1 1)) 5 (3x 1 6, 3y 1 3), so the two compositions do not describe the same transformation.

Not similar; answers may vary. Sample: the fi gures have the same height but different widths so corresponding lengths are not in proportion.

Similar; answers may vary. Sample: dilate the smaller fi gure by some scale factor so it has the same size as the larger fi gure, and then translate onto the larger fi gure.


67

Name Class Date

9-7 Standardized Test Prep Similarity Transformations

Multiple Choice


1. Which similarity transformation maps nABC to nDEF ?

Rx-axis + D0.5 Rx-axis + D2

r(2708, O) + D0.5 r(908, O) + D2

2. Which similarity transformation does not map nPQR to nSTU ?

r(1808, O) + D2 D2 + Rx-axis + Ry-axis

D2 + r(1808, O) D2 + Rx-axis + r(908, O)

3. Which of the following best describes a similarity transformation that maps nJKP to nLMP?

a dilation only

a rotation followed by a dilation

a refl ection followed by a dilation

a translation followed by a dilation

Short Response

4. nABC has vertices A(1, 0), B(2, 4), and C(3, 2). nRST has vertices R(0, 3), S(212, 6), and T(26, 9). What is a similarity transformation that maps nABC to nRST ?

E

FD

A

C

B

x

y

O

R

Q

P

U

T

S

x

y

O

K

J

PM

L

A

I

B

[2] (D3 + r(908, O))(kABC) or equivalent similarity transformation [1] gives one correct transformation of a possible composition of transformations that maps kABC to kRST [0] incorrect or no response


68

Name Class Date

9-7 Enrichment Similarity Transformations

Horizontal and Vertical StretchesIn the coordinate plane, a horizontal stretch with center at the origin and scale factor n maps a point (x, y) to the point (nx, y). Likewise, a vertical stretch with center at the origin and scale factor n maps a point (x, y) to the point (x, ny).

Under horizontal and vertical stretches, images are neither congruent nor similar to the preimage, but certain classifi cations of fi gures can be preserved.

For Exercises 1–3, kABC is the isosceles triangle with vertices A(2a, 0), B(0, b), and C(a, 0).

1. What are the coordinates of the vertices of the triangle after a horizontal stretch with scale factor n?

2. What are the coordinates of the vertices of the triangle after a vertical stretch with scale factor n?

3. Is the image of nABC after a horizontal stretch an isosceles triangle? after a vertical stretch?

4. Suppose the vertices of isosceles triangle nABC are A(0, 0), B(a, b), and C(2a, 0). Are the images of nABC after horizontal or vertical stretches isosceles triangles?

For Exercises 5–8, ABCD is a square with vertices A(2a, 0), B(0, a), C(a, 0), and D(0, 2a).

5. What are the coordinates of the vertices of ABCD after a horizontal stretch with scale factor n?

6. What are the coordinates of the vertices of ABCD after a vertical stretch with scale factor n?

7. Is the image of ABCD after a horizontal stretch a square? after a vertical stretch?

8. What type of quadrilateral is the image of ABCD after a horizontal and vertical stretch?

9. Suppose the vertices of square ABCD are A(0, a), B(a, a), C(a, 0), and D(0, 0). What type of quadrilateral is the image of ABCD after a horizontal or vertical stretch?

A( a, 0) C(a, 0)

B(0, b)y

x

A( a, 0) C(a, 0)

B(0, a)

D(0, a)

y

x

A9(2na, 0), B9(0, b), C9(na, 0)

A9(2a, 0), B9(0, nb), C9(a, 0)

yes; yes

yes

A9(2na, 0), B9(0, a), C9(na,0), D9(0, 2a)

A9(2a, 0), B9(0, na), C9(a,0), D9(0, 2na)

no; no

rhombus

rectangle


69

Name Class Date

9-7 Reteaching Similarity Transformations

Dilations and compositions of dilations and rigid motions form a special class of transformations called similarity transformations. Similarity can be defi ned by similarity transformations as follows.

Two fi gures are similar if and only if there is a similarity transformation that maps one fi gure to the other.

Th us, if you have a similarity transformation, the image of the transformation is similar to the preimage. Use the rules for each transformation in the composition to fi nd the image of a point in the transformation.

Problem

nABC has vertices A(21, 1), B(1, 3), and C(2, 0). What is the image of nABC when you apply the similarity transformation T,22, 25. + D3?

For any point (x, y), D3(x, y) 5 (3x, 3y) and T,22, 25.(x, y) 5 (x 2 2, y 2 5).

(T,22, 25. + D3)(A) 5 T,22, 25.(D3(A)) 5 T,22, 25.(23, 3) 5 (25, 22)

T,22, 25. + D3)(B) 5 T,22, 25.D3(B)) 5 T,22, 25.(3, 9) 5 (1, 4)

(T,22, 25. + D3)(C) 5 T,22, 25.(D3(C)) 5 T,22, 25.(6, 0) 5 (4,25)

Th us, the image has vertices Ar(25, 22), Br(1, 4), and Cr(4, 25) and is similar to nABC .

Exercises

kABC has vertices A(22, 1), B(21, 22), and C(2, 2). For each similarity transformation, draw the image.

1. D2 + Ry-axis 2. r(908, O) + D2

C

B

A

C

B

A x

y

O

x

y

Ox

y

O

B

A

C

A

B

C


70

Name Class Date

9-7 Reteaching Similarity Transformations

If you can show that a similarity transformation maps one fi gure to another, then you have shown that the two fi gures are similar.

Problem

Show that nJKL,nRST by fi nding a similarity transformation that maps one triangle to the other.

nRST appears to be twice the size of nJKL, so dilate by scale factor of 2. Map each vertex using D2.

D2(J) 5 (0, 4) 5 Jr

D2(K) 5 (4, 2) 5 Kr

D2(L) 5 (2, 0) 5 Lr

Graph the image of the dilation nJrKrLr.

nJrKrLr is congruent to nRST and can be mapped to nRST by the glide refl ection Rx-axis + T,25, 0. .

Verify that each vertex of nJrKrLr maps to a vertex of nRST .

(Rx-axis + T,25, 0.)(Jr) 5 Rx-axis(25, 4) 5 (25, 24) 5 R

(Rx-axis + T,25, 0.)(Kr) 5 Rx-axis(21, 2) 5 (21, 22) 5 S

(Rx-axis + T,25, 0.)(Lr) 5 Rx-axis(23, 0) 5 (23, 20) 5 T

Th us, the similarity transformation Rx-axis + T,25, 0. + D2 maps nJKL to nRST .

Exercises

For each pair of fi gures, fi nd a similarity transformation that maps kABC to kFGH . Th en, write the similarity statement.

3. 4.

KJ

L

R

S

T x

y

O

KJ

L

R

S

T x

y

K

J

LO

H

G

F

A

C

By

xO

H

G

F

A

C

By

xO

Answers may vary. Sample: (TR24, 210S + D3)(kABC) 5kFGH

Answers may vary. Sample: (r(2708, O) + D1.5)(kABC) 5 kFGH


71

Name Class Date

Chapter 9 Quiz 1 Form G

Lessons 9-1 through 9-4

Do you know HOW?


1. 2.

3. If a transformation maps GHIJ to GrHrIrJr, what is the image of I? What is the

image of GH ?

4. Point R(x, y) moves 13 units right and 14 units down. What is a rule that describes this translation?

Draw the image of each fi gure for the given transformation.

5. Rx-axis(ABCD) 6. r(908, O)(ABCD)

7. nONM has vertices O(24, 2), N(3, 6), and M(0, 3). What are the coordinates of the vertices of (Ry51 + T,3, 0.)(nONM)

8. Is the transformation of AB with vertices A(2, 3) and B(21, 2), fi rst across x 5 4, and then across y 5 22, a translation or a rotation? For a translation, describe the direction and distance. For a rotation, tell the center of rotation and the angle of rotation.

Do you UNDERSTAND?

9. Reasoning Th e point (21, 21) is the image under the translation T,25, 5.(x, y). What is its preimage?

10. Reasoning In the diagram, is one fi gure a refl ection image, a translation image, or a rotation image of the other?

Image

Preimage ImagePreimage

x

A

D C

B

y

2

4 xO

y

2

2

2

A

D C

B

Yes; preserves distance and angle measure

l9; G9H9

TR13, 214S(R)

(4, 26)

rotation image

No; does not preserve distance

D

A B

C B

A D

C

(21, 0), (6, 24), (3, 21)

rotation around (4, 22); 1808


72

Name Class Date

Chapter 9 Quiz 2 Form G


Do you know HOW?

Write a congruence or similarity statement for the two fi gures in each coordinate grid. Th en write a congruence transformation or similarity transformation that maps one fi gure to the other.

1. 2.

3. 4.

5. Th e solid-line fi gure is a dilation of the dashed-line fi gure with center of dilation M. Is the dilation an enlargement or a reduction? What is the scale factor of the dilation?

6. Given P(3, 11), what are the coordinates of D3(P)?


Do you UNDERSTAND? 8. Compare and Contrast Explain how a similarity transformation is the same

as or diff erent from a congruence transformation.

9. Error Analysis A student used image and preimage lengths of an enlargement to fi nd a scale factor of 34. Explain her possible error.

G

HF

C

B

Ax

y

O

P

RQ

F

E

Dx

y

U

TS

N

ML

x

y

O

C

B

A

N

ML

x

y

O

M

7 10

In a similarity transformation, the image and the preimage are similar but not usually congruent. In a congruence transformation, the preimage and image are congruent. Similarity transformations are compositions of rigid motions and dilations, while congruence transformations are compositions of rigid motions.

An enlargement must have a scale factor greater than 1. The image length should be the numerator and the preimage length should be the denominator.

kABC O kFGH; Answers may vary.Sample: (Rx-axis + TR3, 21S)(kABC ) 5 kFGH

kLMN,kSTU; Answers may vary.Sample: (T

R4,24S + D0.5)(kLMN ) 5 kSTU

kDEF O kPQR; Answers may vary.Sample: (T

R1,23S + r(908, O))(kDEF ) 5 kPQR

kABC,kLNM; Answers may vary.Sample: (Rx-axis + D2)(kABC ) 5 kLNM

enlargement; 107

(9, 33)

(8, 216)


73

Name Class Date

Chapter 9 Test Form G

Do you know HOW?

1. nRrSrTr is a translation image of nRST . What is a rule for the translation?

2. Is a glide refl ection a rigid motion? Explain.

Find the coordinates of the vertices of each image.

3. Ry-axis(QRST)

4. r(2708, O)(QRST)

5. D5(QRST)

6. T,2,25.(QRST)

7. (Ry522 + T,24, 0.)(QRST)

Write a single transformation rule that has the same eff ect as each composition of transformations.

8. T,6, 7. + T,210, 7.

9. Rx56 + Rx51

10. Ry52 + T,0, 1.

Identify the rigid motion that maps the solid-line fi gure onto the dotted-line fi gure.

11. 12.

Find the image of each point for the given dilation.

13. L(6,4); D1.2(L) 14. S(22,26); D0.25(S) 15. W(23, 2); D5(W)

x

R S

T

SR

T

y

2

2O 4 624

4

4

2

x

Q

T

R

y

2

S 4 624

4

2

T,27, 3S(x, y)

Yes; a glide refl ection preserves distance and angle measure because it is a composition of translation and refl ection.

Q9(21, 5), R9(23, 21), S9(0, 0), T9(2, 3)

Q9(3, 0), R9(5, 26), S9(2, 25), T9(0, 22)

refl ectiontranslation

(7.2, 4.8) (20.5, 21.5) (215, 10)

Q9(23, 29), R9(21, 23), S9(24, 24), T9(26, 27)

Q9(5, 21), R9(21, 23), S9(0, 0), T9(3, 2)

Q9(5, 25), R9(15, 25), S9(0, 0), T9(210, 15)

TR24, 14S

TR10, 0S

Ry51.5


74

Name Class Date


16. 17.

18. 19.

Do you UNDERSTAND?

20. Reasoning Lines a and b are parallel. Th e letter A is refl ected fi rst across line a so that the image is between lines a and b. Th e fi gure is then refl ected across line b. Describe the image of A.

21. Vocabulary What is the image of (26, 6) after a refl ection across the y-axis?

22. Coordinate Geometry nABC has vertices at A(0, 5), B(4, 4), and C(21, 0). Graph the image T,21, 22.(nABC).

B

C

A

Q

P

R x

y

O

V U

W

N

ML

x

y

O

JK

L

I

H

G x

y

O

MK

L

H

G

F x

y

O

Chapter 9 Test (continued) Form G

This series of refl ections is a translation. The image of A is the same size and in the same orientation as the original, but it is translated twice the distance between lines a and b.

(6, 6)

x

AB

C

y

2

2 44

4

4

2

kABC O kPQR; Answers may vary.Sample: (r(908, O) + Ry-axis)(kABC) 5 kPQR

kGHI 5 kJKL; Answers may vary.Sample: (Rx-axis + r(2708, O))(kGHI) 5 kJKL

kLMN,kUVW ; Answers may vary.Sample: (r(1808, O) + D2)(kLMN) 5 kUVW

kFGH O kKLM; Answers may vary.Sample: (T

R1, 25S + Ry-axis)(kFGH) 5 kKLM


75

Name Class Date

Chapter 9 Quiz 1 Form K


Do you know HOW?


1. 2.

3. If a transformation maps nABC to nArBrCr, what is the image of A?

4. If a transformation maps nABC to nArBrCr, what is the image of BC?

5. Point R(x, y) moves 7 units left and 3 units up. What is a rule that describes this translation?

Draw the image of each fi gure for the given transformation.

6. T,3, 2.(nABC) 7. Ry-axis(nABC)

8. nXYZ has vertices X(22, 2), Y(21, 5), and Z(2, 3). What are the coordinates of the vertices of (Rx53 + T,0, 23.)(nXYZ)?

Do you UNDERSTAND?

9. Reasoning Th e point (1, 1) is the image under the translation T,3, 23. . What is the preimage of this point?

10. Reasoning In the diagram, is one fi gure a refl ection image, a translation image, or a rotation image of the other?

A

B C C

D

A

E F

B

D

EF

B

A

CD

AB

C D

A

BC

y

O

4

4

2

x

B

C

A

O

4

2

2 4

2

2

y

x

X

YZ

2

2

4

4 6 8

y

xO2

yes; preserves distance and angle measure no; does not preserve distance

A9

B9C 9

TR27, 3S(R)

(22, 4)

rotation image

A

BC

B

C

(8,21), (7, 2), (4, 0)


76

Name Class Date

Chapter 9 Quiz 2 Form K


Do you know HOW?


1. 2.

3. 4.

5. Th e solid-line fi gure is a dilation of the dashed-line fi gure with center of dilation D. Is the dilation an enlargement or a reduction? What is the scale factor of the dilation?


7. Given P(10, 220), what are the coordinates of D0.2(P)?

Do you UNDERSTAND? 8. Compare and Contrast How is a similarity transformation diff erent from an

a congruence transformation? How are they similar?

9. Error Analysis A student uses image and preimage lengths of a reduction to fi nd a scale factor of 1.5. Explain how you know she has made an error.

F

G

HC

B

A x

y

O Q

P R

KJ

Lx

y

O

S

U

T

ED

Fx

y

O

F D

E

BA

Cx

y

O

D

8

2

A similarity transformation maps a fi gure to a similar but not congruent fi gure. A congruence transformation maps a fi gure to a congruent fi gure both are compositions of transformations. A similarity transformation is composition of dilations and rigid motions, while a congruence transformation is composition of just rigid motions.

kABC O kFGH; Answers may vary. Sample: (Rx-axis + T

R5, 0S)(kABC) 5 kFGH

kDEF,kSTU; Answers may vary. Sample: (T

R0, 24S + D2)(kDEF ) 5 kSTU

kJKL O kPQR; Answers may vary. Sample: (T

R3, 0S + r(908, O))(kJKL) 5 kPQR

kABC O kDEF ; Answers may vary. Sample: (r(908, O) + Rx-axis)(kABC)5 kDEF

reduction; 14

(10, 25)

(2, 24)

The scale factor R 1; she may have divided the longer length by the shorter length.


77

Name Class Date

Chapter 9 Test Form K

Do you know HOW?

1. In the fi gure at the right, nArBrCr is a translation image of nABC. What is a rule for the translation?

2. Is a rotation a rigid motion? Explain.

Find the coordinates of the vertices of each image.

3. T,21, 2.(MATH)

4. Rx-axis(MATH)

5. r(908, O)(MATH)

6. D2(MATH)

7. (Ry-axis + T,0, 2.)(MATH)

Write a single transformation rule that has the same eff ect as each composition of transformations on the given fi gure.

8. T,23, 4. + T,1, 22.

9. Ry522 + Ry52

10. Rx51 + Ry521

Identify the rigid motion that maps the solid-line fi gure to the dashed-line fi gure.

11. 12.

A

B

C

A

xO

y

B

C

2

2

2

4

2

4

4 6 8

H

T

A

x

y

M

O 2

2

24

2

4

4

M

P

N

x

y

O 2

2

4

24

2

4

4

TR25, 6S(x, y)

M9(23, 4), A9(2, 2), T9(21, 22), H9(25, 0)

M9(22, 22), A9(3, 0), T9(0, 4), H9(24, 2)

M9(22, 22), A9(0, 3), T9(4, 0), H9(2, 24)

M9(24, 4), A9(6, 0), T9(0, 28), H9(28, 24)

M9(2, 4), A9(23, 2), T9(0, 22), H9(4, 0)

TR0, 28S

r(1808, (1, 21))

TR22, 2S

refl ectiontranslation

Yes; a rotation preserves distance and angle measure, so it is a rigid motion.


78

Name Class Date

Chapter 9 Test (continued) Form K


13. 14.

15. 16.

Find the image of each point for the given dilation.

17. A(5, 210); D0.4(A) 18. B(21, 22); D3(B) 19. C(4, 6); D2.5(C)

Do you UNDERSTAND?

20. Coordinate Geometry nXYZ has vertices at X(22, 3), Y(3, 0), and Z(2, 2). Graph the image of nXYZ for the translationT,3, 1.(nXYZ).

21. Reasoning A transformation maps each point (x, y) to (2kx, ky). Write this transformation as a composition of a dilation and rigid motion.

22. Vocabulary What is the image of (2, 3) after a refl ection across the y-axis?

FE

D

B

AC x

y

O

L

K

J

G

I

Hx

y

O

S

R

Q

NM

P x

y

O V

UTX

Z

Yx

y

O

ZX

Y

xO

y

2

2

4

2 4 6 8

4

XZ

Y

kABC O kDFE; Answers may vary. Sample: (r(908, O) + Ry-axis)(kABC) 5 kDFE

kMNP O kQRS; Answers may vary.Sample: (T

R25, 0S + r(2708, O))(kMNP) 5 kQRS

Ry-axis + Dk

kGHI,kJKL; Answers may vary. Sample: D0.5(kGHI) 5 kJKL

kTUV,kXYZ ; Answers may vary.Sample: (T

R24, 23S + D2)(kTUV ) 5 kXYZ

(2, 24) (23, 26) (10, 15)

(22, 3)


79

Name Class Date

Performance Tasks

Chapter 9

Task 1A flea is 0.2 cm long. As it walks across the glass surface of an overhead projector, its image on the screen appears to be 4 cm long. What is the scale factor of the dilation? If the flea jumps 10 cm across the glass surface, how far does its image jump across the screen? Explain your reasoning.

Task 2A composition of transformations used to describe a congruence transformation is not unique. For example, the composition r(908, O) + T,5, 2. maps the point (x, y) to (2y 2 2, x 1 5), as does the composition T,22, 5. + r(908, O). Write another example of transformation composed of a translation followed by a 90° rotation, and then write an equivalent transformation composed of a 90° rotation followed by a translation. Show that each point (x, y) maps to the same point under both transformations.

Task 3In the real world, not all transformations are always possible. For example, if you drive a car around town, what types of transformations does the car undergo? What types can it not undergo? Explain.

20; 200 cm; all distances are magnifi ed 20 times. [4] Student gives correct answers and an accurate explanation. [3] Two out of three correct answers are given; student used the correct strategy and gives an accurate explanation. [2] Student gives answers and an explanation that are largely correct. [1] Student gives answers or an explanation with serious errors. [0] Student makes little or no effort.

Check students’ work. [4] Student gives a correct example of two equivalent compositions and shows all work. [3] Student gives a correct example of two equivalent compositions but shows little work. [2] Student gives a correct example of two equivalent compositions but shows no work. [1] Student gives an example of transformations with serious errors. [0] Student makes little or no effort.

Answers may vary. Sample: The car can undergo translations and rotations as it moves and turns. But it cannot undergo a dilation, which would change its size, nor a refl ection, which would, for example, put the steering wheel on the passenger side. [4] All correct answers (possible: translation, rotation; not possible: refl ection, dilation) and an accurate explanation are given. [3] Three out of four correct answers and an accurate explanation are given. [2] Two out of four correct answers and an explanation that is largely correct are given. [1] Student gives answers or an explanation with serious errors. [0] Student makes little or no effort.


80

Name Class Date

Task 4Draw a quadrilateral in the coordinate grid at the right. Show the fi gure undergoing the transformation rules you have studied in this chapter. Use diff erent coordinate grids for each transformation. Draw each image, and write the rule you used.

a. translation b. dilation

c. refl ection d. glide refl ection

e. rotation

x

y

2

4

4

4 4

2

4

2

2

4

2

4 4

y

xO 2 4

4

2

24

2

4

y

xO

2

2

4

2

2

4

4 4

y

xO 2

2

4

2

2

4

4 4

y

xO

4

2

422

4

2

4

y

xO

Performance Tasks (continued)

Chapter 9

Ry522 + TR2, 0S

Check students’ work. [4] Student provides correct transformation rules. [3] Student provides three or four out of fi ve correct transformation rules. [2] Student provides two out of fi ve correct transformation rules. [1] Student provides transformation rules with serious errors. [0] Student makes little or no effort.

r(908, O)

TR1, 0S

D2

Ry51


81

Name Class Date

Cumulative Review

Chapters 1–9

Multiple Choice

1. A woman has a piece of wood that is 22 ft long and another that is 13 ft long. She wants to select another piece of wood so that she can put all the pieces together to make a triangular garden bed. How long could the third piece of wood be?

8 ft 8 ft 6 in. 12 ft 36 ft

2. A scientist looks at small pond organisms using a microscope. In his view, one of the organisms is 2.2 cm long when magnifi ed to 100 times its actual size. What is its actual length, in mm?

0.022 mm 0.22 mm 22 mm 220 mm

3. For what values of x and y are the triangles congruent?

x 5 3, y 5 12 x 5 12, y 5 3

x 5 4, y 5 1 x 5 3, y 5 10

4. Th e lengths of the sides of a triangle are in the extended ratio 3i7 i9. Th e triangle’s perimeter is 228 m. What are the lengths of the sides?

30, 70, and 90 m 36, 84, and 108 m

33, 77, and 107 m 37, 84, and 111 m

5. A painter leans a ladder up against a house. Th e ladder is 12 ft long and the base of the ladder is 3.4 ft away from the house. Th e ladder reaches the base of a window. To the nearest tenth of a foot, how high off the ground is the base of the window?

8.6 ft 11.5 ft 11.6 ft 12.5 ft

Short Response 6. A game comes with directions to cut out a fi gure from paper

and fold it into a three-dimensional fi gure. What type of fi gure will be formed using the net shown?

7. Hexagon HEXAGZ has vertices H(3, 0), E(1, 22), X(21, 22), A(23, 0), G(22, 2), and Z(1, 2). What are the coordinates of the vertices of D3.5(HEXAGZ)?

3x y 3

y

18

2y 4x

C

H9(10.5, 0), E9(3.5, 27), X9(23.5, 27), A9(210.5, 0), G9(27, 7), Z9(3.5, 7)

G

A

H

B

pentagonal pyramid[2] correct answer [1] some type of pyramid [0] incorrect answer

[2] all six vertices correct [1] at least three vertices correct [0] two or fewer vertices correct


82

Name Class Date

Cumulative Review (continued)

Chapters 1–9

Short Response

8. In nABC, AB 5 30, AC 5 35, and m/A 5 728. To the nearest tenth, what are the unknown measures of the triangle?

9. nABC has vertices A(22, 0), B(0, 4), and C(1, 2). What are the coordinates of the vertices of (r(908, O) + Rx-axis)(nABC)?

10. Point T is the centroid of the triangle.If AZ 5 12.36, what is AT? What is TZ?

11. A ramp leading to a stage forms a 378 angle with the fl oor. If the ramp is 22 ft long, how high is the stage from the ground? Round to the nearest inch.

12. Th e coordinates of the endpoints of AB are A(15, 6) and B(3, 9). Find the

length of AB to the nearest tenth. What is the midpoint of AB?

Gridded Response

13. Th e distance from one corner of a square tablecloth to the opposite corner is 102 in. To the nearest foot, what is the side length of the tablecloth?

14. What is the measure of /x?

A

C BZ

X YT

x

119

Answers

13. 14.

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

9876543210

AT 5 8.24; TZ 5 4.12[2] both answers correct [1] one answer correct [0] no answers correct

159 in. or 13 ft 3 in. [2] correct answer [1] answer rounded incorrectly [0] incorrect answer

12.4; (9, 7.5)[2] both answers correct [1] one answer correct [0] no answers correct

6 6 1

BC N 38.4, mlB N 60.18, mlc N 48.08 [2] all three measures are correct [1] at least BC is correct [0] no answers correct

A9(0, 22), B9(4, 0), C 9(2,1) [2] all three vertices are correct [1] at least one measurement is correct [0] no answers correct


83

T E A C H E R I N S T R U C T I O N S

Chapter 9 Project Teacher Notes: Frieze Frames

About the ProjectStudents will identify frieze patterns that were produced using refl ections, translations, rotations, and glide refl ections. Students will learn how to classify the patterns by their symmetries. Th ey will also design their own patterns.

Introducing the Project• Have each team draw a nonregular polygon, trace it, and cut it out. Th en have

the teams use their cutouts to make patterns.

• Have students try to copy a shape by drawing in a mirror.

Activity 1: InvestigatingAsk students to describe wallpaper borders or stencils they have seen that have frieze pattems. Discuss how the patterns are repeated. You may want to have students look through department store catalogs, home-decorating magazines, or wallpaper books for examples. Students must be familiar with translations and refl ections.

Activity 2: ClassifyingStudents also may want to classify the patterns from the previous activities.

Activity 3: DesigningStudents may want to divide the work by having each group member make a design for one of the types. Th en the group can complete the remaining types together.

Finishing the ProjectYou may wish to plan a project day on which students share their completed projects. Encourage students to share their processes as well as their products.

• Have students review their sketches, classifi cations, and design needed for the project.

• Ask students to share any insights they gained when completing the project, such as how they identifi ed diff erent transformations.


84

Name Class Date

Chapter 9 Project: Frieze Frames

Beginning the Chapter Project Ukrainian painted eggs, dollar bills, Native American pottery, Japanese kimonos, automobile tire treads, and African cloths are products of vastly diverse cultures, but all these things have something in common. Th ey contain strips of repeating patterns, called frieze patterns.

In this chapter project, you will explore the underlying relationships among frieze patterns from around the world. You also will produce your own designs. You will see how distinct civilizations—separated by oceans and centuries—are linked by their use of geometry to express themselves and to beautify their world.

Activities

Activity 1: InvestigatingA frieze pattern, or strip pattern, is a design that repeats itself along a straight line. Every frieze pattern can be mapped onto itself by a translation. Some also can be mapped onto themselves by other transformations. Decide whether each frieze pattern can be mapped onto itself by a refl ection, a rotation, a translation, or a glide refl ection.

a. Nigerian design b. Ancient Egyptian ornament c. Arabian design

Activity 2: ClassifyingIt may surprise you to fi nd out that when you classify frieze patterns by their symmetries, there are only seven diff erent types. Each pattern is identifi ed by a diff erent two-character code: 11, 1g, m1, 12, mg, 1m, or mm. Use the fl owchart at the right to classify each frieze pattern below.

a. Caucasian rug design, Kazakh

b. French, Empire motif

NoDoes the strip have vertical line symmetry?

First symbol is m.

Second symbol is m.

Second symbol is g.

Second symbol is 2.

Yes

Yes

Yes

Yes

First symbol is 1.

Does the strip have a horizontal line of refl ection?

Is there a glide refl ection?

Is there a half-turn?

Second symbol is 1.

No

No

No

refl ection and translation

mg

12

rotation and translation

refl ection, rotation, translation, and glide refl ection


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Name Class Date

Chapter 9 Project: Frieze Frames (continued)

Activity 3: DesigningIn previous activities, you investigated and classifi ed frieze patterns from a variety of cultures. Now you can make your own. Use graph paper, dot paper, geometry or drawing software, or cutouts. Make at least one frieze pattern for each of the seven types summarized at the right.

Finishing the ProjectUsing one or more of the techniques you explored, make your own design. Prepare a frieze pattern display. Include a brief explanation of frieze patterns as well as your original designs for each of the seven types of frieze patterns. Find and classify more examples of frieze patterns from various cultures, clothing, and other places in your home, school, and community.

Refl ect and ReviseAsk a classmate to review your project with you. Together, check that the diagrams and explanations are clear and accurate. Revise your work as needed. Consider doing more research.

Extending the ProjectUse logical reasoning and what you’ve learned about transformations to explain why there are no more than seven diff erent frieze patterns. List all other possible combinations of symmetries, and show how each can be ruled out.

Th e Seven Types of Frieze Patterns

Types Symmetries Examples

11 T P P P P P P P

12 T, H Z Z Z Z Z Z Z

m1 T, RV Y Y Y Y Y Y Y

1g T, G D

M

D M D

M

D

1m T, RH, G D D D D D D D

mg T, H, RV, G M

M

M

M

M

M

M

mm T, H, RV, RH, G I I I I I I I

Key to Symmetries:T 5 TranslationH 5 Half-turnRV 5 refl ection in vertical line

RH 5 Refl ection in horizontal lineG 5 Glide refl ection


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Chapter 9 Project Manager: Frieze Frames

Getting StartedRead about the project. As you work on it, you will need several sheets of graph paper or dot paper. If available, you may use geometry software or drawing software also. Keep all your work for the project in a folder, along with this Project Manager.

Checklist Suggestions

☐ Activity 1: transformations

☐ Review refl ections, rotations, and glide refl ections before completing this activity.

☐ Activity 2: seven types of friezes

☐ Run through the fl owchart twice to be sure.

☐ Activity 3: pattern design

☐ Check your work by reviewing it with a friend.

☐ Frieze Pattern display ☐ Use a map or time line to add interest to your display.

Scoring Rubric

4 Your information is accurate and complete. Your diagrams and explanations are clear. You use geometric language appropriately and correctly. You have included many frieze patterns exemplifying a wide variety of styles, cultures, media, and geometric properties. Your display is organized and complete.

3 Your information may contain a few minor errors. Your diagrams and explanations are understandable. Most of the geometric language is used appropriately and correctly. You have included a moderate number of frieze patterns of some variety. Your display shows some organization.

2 Much of the information is incorrect. Diagrams and explanations are hard to follow or misleading. Geometric terms are completely lacking, used sparsely, or often misused. Th e display does not include many examples and shows little if any attempt at organization.

1 Major elements of the project are incomplete or missing.

0 Project not handed in, or work does not follow instructions.

Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.

Teacher’s Evaluation of Project

Documents

Name Class Date 9-1 - Pequannock Township High Schoolhs.pequannock.org/ourpages/auto/2014/8/26/59171599... · Name Class Date 9-1 Additional Vocabulary Support Translations Choose