9-1 translations - trimble county schools · 9-1 translations review 1. ... cross out the word that...

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Vocabulary

Chapter 9 222

9-1 Translations

Review

1. Underline the correct word to complete the sentence.

A transformation of a geometric figure is a change in the position, shape, or

color / size of the figure.

2. Cross out the word that does NOT describe a transformation.

erase flip rotate slide turn

Vocabulary Builder

isometry (noun) eye SAHM uh tree

Definition: An isometry is a transformation in which the preimage and the image of a geometric fi gure are congruent.

Example:

Preimage Image

Non-Example:

Preimage Image

Use Your Vocabulary

Complete each statement with congruent, image or preimage.

3. In an isometry of a triangle, each side of the 9 is congruent to each side of the preimage.

4. In an isometry of a trapezoid, each angle of the image is congruent to each angle of the 9.

5. An isometry maps a preimage onto a(n) 9 image.

image

preimage

congruent

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d.Problem 1

Problem 2

Preimage Image

U

N

S

PI

D

AA BB CC

B

A

C

B

AC

223 Lesson 9-1

Identifying an Isometry

Got It? Does the transformation below appear to be an isometry? Explain.

6. Name the polygon that is the preimage. 7. Name the polygon that is the image.

8. Do the preimage and image appear congruent? Yes / No

9. Does the transformation appear to be an isometry? Explain.

_______________________________________________________________________

_______________________________________________________________________

Naming Images and Corresponding Parts

Got It? In the diagram, kNIDmkSUP. What are the images of lI and point D?

10. The arrow ( S ) shows that n is the image of nNID,

so nNID > n .

11. Describe how to list corresponding parts of the preimage and image.

_______________________________________________________________________

_______________________________________________________________________

12. Circle the image of /I .

/I /S /P /U

13. Circle the image of point D.

I S P U

Key Concept Translation

A translation is a transformation that maps all points of a fi gure the same distance in the same direction.

A translation is an isometry. Prime notation ( r) identifi es image points.

14. If ~PQRS is translated right 2 units, then every point on

~PrQrRrSr is units to the right of its preimage point.

kite kite

SUP

SUP

Yes. Explanations may vary. Sample: The preimage and the image

appear congruent, so the transformation appears to be an isometry.

List corresponding parts of the preimage and image in the same order.

2

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Problem 4

Problem 3

xyO2

3

46L

M

N

Chapter 9 224

Finding the Image of a Translation

Got It? What are the images of the vertices of kABC for the translation (x, y)m (x 1 1, y 2 4)? Graph the image of kABC .

15. Identify the coordinates of each vertex.

A( , )

B( , )

C( , )

16. Use the translation rule (x, y) S (x 1 1, y 2 4) to find Ar, Br, and Cr .

Ar( 1 1, 2 4) 5 Ar( , )

Br( 1 1, 2 4) 5 Br( , )

Cr( 1 1, 2 4) 5 Cr( , )

17. Circle how each point is translated.

1 unit to the right and 4 units up 1 unit to the right and 4 units down

1 unit to the left and 4 units up 1 unit to the left and 4 units down

18. Graph the image of nABC on the coordinate plane above.

Writing a Rule to Describe a Translation

Got It? The translation image of kLMN is kL9M9N9 with L9(1, 22), M9(3, 24), and N9(6, 22). What is a rule that describes the translation?

19. Circle the coordinates of point L.

(6, 21) (21, 26) (26, 21) (21, 6)

20. Circle the coordinates of point M.

(24, 23) (23, 24) (24, 3) (23, 4)

21. Circle the coordinates of point N.

(21, 1) (1, 21) (21, 0) (21, 21)

22. Find the horizontal change from L to L’. 23. Find the vertical change from L to L’.

1 2 5 22 2 5

Underline the correct word to complete each sentence.

24. From nLMN to nLrMrNr, each value of x increases / decreases .

25. From nLMN to nLrMrNr, each value of y increases / decreases .

26. A rule that describes the translation is? (x, y) S ( , ).

x

y

O 42

2

2

2

4

6

4

A

A’B’

C’

C

B

22

22

1

0 –1 1 25

1 2 23

2 21 22

2

11

0 21

x 1 7

26 7 21 21

y 2 1

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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

Problem 5

A C

B

PR

Q

2

1

225 Lesson 9-1

• Do you UNDERSTAND?

Error Analysis Your friend says the transformation kABC SkPQR is a translation. Explain and correct her error.

31. Find the distance between the preimage and image of each vertex.

BQ 5 AP 5 CR 5

32. Does this transformation map all points the same distance? Yes / No

33. Is nABC SnPQR a translation? Explain.

_______________________________________________________________________

34. Correct your friend’s error.

_______________________________________________________________________

Composing Translations

Got It? The diagram at the right shows a chess game with the black bishop 6 squares right and 2 squares down from its original position after two moves. The bishop next moves 3 squares left and 3 squares down. Where is the bishop in relation to its original position?

27. If (0, 0) represents the bishop’s original position, the bishop is now

at the point ( , ).

28. Write the translation rule that represents the bishop’s next move.

(x, y) S (x 2 , y 2 )

29. Substitute the point you found in Exercise 27 into the rule you wrote in Exercise 28.

( , ) u ( 2 , 2 )

30. In relation to (0, 0), the bishop is at ( , ).

Check off the vocabulary words that you understand.

transformation preimage image isometry translation

Rate how well you can fi nd transformation images.

6

3

6 22

5 9 1

22

No. The distances between preimage and image vertices are not equal.

Answers may vary. Sample:

The transformation is a flip.

3

6 5 3 3

3 25

Explanations may vary. Sample:

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Vocabulary

Review

The preimage B andits image B’ areequidistant fromthe line of reflection.

B

C

A

r

C

B

A

Chapter 9 226

9-2 Reflections

1. Circle the translation rule that shows a mapping 2 units left and 1 unit up. Underline the translation rule that shows a mapping 2 units right and 1 unit down.

(x, y) S (x 2 2, y 1 1) (x, y) S (x 1 2, y 2 1) (x, y) S (x 2 2, y 2 1)

Vocabulary Builder

refl ection (noun) rih FLEK shun

Related Words: line of reflection

Definition: A reflection is a mirror image of an object that has the same size and shape but an opposite orientation.

Math Usage: A reflection is a transformation where each point on the preimage is the same distance from the line of refl ection as its reflection image.

Use Your Vocabulary

Write T for true or F for false.

2. A reflection is the same shape as the original figure.

3. A reflection makes a figure larger.

Key Concept Reflection Across a Line

Refl ection across a line r, called the line of refl ection, is a transformation with these two properties:

• If a point A is on line r, then the image of A is itself (that is, A r 5 A).

• If a point B is not on line r, then r is the perpendicular bisector of BBr .

A refl ection across a line is an isometry.

4. Line is the perpendicular bisector of CCr.

T

F

r

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d.Problem 1

Problem 2

x

y

O 42

2

4

6

24

PP’

x

y

O 42

2

4

2

2

4

4

A

C

B

A’

B’

C’

227 Lesson 9-2

Reflecting a Point Across a Line

Got It? What is the image of P(3, 4) reflected across the line x 5 21?

5. Graph P on the coordinate plane at the right.

6. Describe the line of reflection. Then graph the line of reflection.

_______________________________________________

_______________________________________________

7. The distance from point P to the line of reflection is units.

Underline the correct word(s) to complete each sentence.

8. The x-coordinates of P and Pr are different / the same .

9. The y-coordinates of P and Pr are different / the same .

10. Point P is reflected to the left / right across the line of reflection.

11. Graph the image of P(3, 4) and label it Pr .

12. The coordinates of Pr are ( , ).

Graphing a Reflection Image

Got It? Graph points A(23, 4), B(0, 1), and C(4, 2). What is the image of kABC reflected across the x-axis?

13. The x-axis is the line y 5 .

14. Circle the distance in units from point A to the x-axis. Underline the distance from point B to the x-axis. Put a square around the distance from point C to the x-axis.

0 1 2 3 4

15. Point Br is unit(s) below the x-axis.

16. Point Cr is unit(s) below the x-axis.

17. The arrow shows how to find vertex Ar . Graph the image of nABC and label vertices Br and Cr on the coordinate plane below.

Answers may vary. Sample: x 5 21 is

0

1

2

a vertical line.

4

4–5

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Problem 3

R

t

O

P

R’

O

R

P

tO’

Chapter 9 228

Minimizing a Distance

Got It? Reasoning The diagram shows one solution of the problem below. Your classmate began to solve the problem by reflecting point R across line t. Will her method work? Explain.

Beginning from a point on Summit Trail (line t), a hiking club will build a trail to the Overlook (point O) and a trail to Balance Rock (point R). The club members want to minimize the total length of the two trails. How can you find the point on Summit Trail where the two new trails should start?

You need to fi nd the point P on line t such that the distance OP 1 PR is as small as possible. In the diagram, the problem was solved by locating Or, the refl ection image of O across t. Because t is the perpendicular bisector of OOr, PO 5 POr, and OP 1 PR 5 OrP 1 PR. By the Triangle Inequality Th eorem, the sum OrP 1 PR is least when R, P, and Or are collinear. So, the trails should start at the point P where ROr intersects line t.

Place a ✓ in the box if the response is correct. Place an ✗ if it is incorrect.

18. When point R is reflected across line t, t is the perpendicular bisector of RRr.

19. PR 2 PRr

20. RP 1 PO 5 RrP 1 PO

21. Points O, P, and Rr are NOT collinear.

22. The trails should start at the point P where ORr intersects t.

23. Reflect R across line t in the diagram at the right. Label the reflection Rr .

24. Draw RRr .

25. Draw RrO.

26. Label the point where RrO intersects line t

as point P. Draw PR.

27. What do you notice about point P after reflecting R across line t?

_______________________________________________________________________

_______________________________________________________________________

28. Will your classmate’s method work? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

✓

✓

✓

✗

✗

Answers may vary. Sample: Reflecting point R across line t locates the

Yes. Explanations may vary. Sample: Her method works because

both methods locate the point at the same place on line t.

same point P as found in the solution shown in the diagram.

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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

x

y

O 42

2

4

2

2

4

4

P’

P”

P

229 Lesson 9-2


What are the coordinates of a point P(x, y) reflected across the y-axis? Across the x-axis?

29. Reflect point P across the y-axis. Label the image Pr .

30. Circle the coordinates of point P.

(3, 1) (23, 21) (23, 1) (3,21)

31. Circle the coordinates of point Pr .

(3, 1) (23, 21) (23, 1) (3, 21)

32. Describe how the coordinates of Pr are different from the coordinates of P.

_______________________________________________________________________

_______________________________________________________________________

33. Reflect point P across the x-axis. Label the image Ps . The coordinates of Ps

are ( , ).

34. Describe how the coordinates of Ps are different from the coordinates of P.

_______________________________________________________________________

35. Complete the model below to find the coordinates of P(x, y) reflected across the y-axis and across the x-axis.


reflection line of reflection

Rate how well you can fi nd refl ection images of fi gures.

Reflected across y-axis.

Reflected across x-axis.

x y,

x y,

Preimage

(x, y)

( )

( )

The x-coordinates are opposites. The y-coordinates do not change.

The x-coordinates do not change. The y-coordinates are opposites.

23 21

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Vocabulary

x

y

O 4 6

2

246 2

AA’

Chapter 9 230

9-3 Rotations

Review

1. The diagram at the right shows the reflection of point A across a line of reflection. Draw the line of reflection.

2. Circle the equation of the line of reflection in the diagram above.

x 5 1 y 5 1 x 5 2 y 5 2

Vocabulary Builder

rotation (noun) roh TAY shun

Definition: A rotation is a spinning motion that turns a fi gure about a point or a line.

Related Words: center of rotation, axis of rotation

Math Usage: A rotation about a point is a transformation that turns a fi gure clockwise or counterclockwise a given number of degrees.

Use Your Vocabulary

Complete each statement with always, sometimes, or never.

3. The rotation of the moon about Earth 9 takes a year.

4. A rotation image 9 has the same orientation as the preimage.

5. A transformation is 9 a rotation.

6. A rotation is 9 a transformation.

7. A 1108counterclockwise rotation is the same as a 2508 clockwise rotation about the same point.

rotation

never

always

always

always

sometimes

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Problem 1

V

R

R

W

U

U

WV

x

V UU

WV

The preimage V andits image V areequidistant fromthe center of rotation.

L

O

B

L

L

O

O´

B

231 Lesson 9-3

Drawing a Rotation Image

Got It? What is the image of kLOB for a 508 rotation about B?

11. Describe the image of B.

_______________________________________________________________________

_______________________________________________________________________

12. Follow the steps below to draw a rotation image.

Step 1 Use a protractor to draw a 508 counterclockwise angle with vertex B and side BO.

Step 2 Use a compass to construct BOr > BO.

Step 3 Use a protractor to draw a 508 angle with vertex B and side BL.

Step 4 Use a compass to construct BLr > BL.

Step 5 Draw nLrOrBr .

Key Concept Rotation About a Point

A rotation of x8 about a point R, called the center of rotation, is a transformation with these two properties:

• Th e image of R is itself (that is, Rr 5 ).

• For any other point V, RVr 5 RV and m/VRVr 5 x.

Th e positive number of degrees a fi gure rotates is the angle of rotation.

A rotation about a point is an isometry.

Use the diagram above for Exercises 8–10.

8. The preimage is n and the image is n .

9. RWr 5 and m/WRWr 5 .

10. RUr 5 and m/URUr 5 .

R

UVW U9V9W9

RW

RU

x

x

Answers may vary. Sample: The image of B is itself, so B9 5 B.

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Problem 3

Problem 2

CM

Q

X

Chapter 9 232

Th e center of a regular polygon is the point that is equidistant from its vertices. Th e center and the vertices of a regular n-gon determine n congruent triangles.

13. The center and the vertices of a square determine congruent triangles.

Identifying a Rotation Image

Got It? Point X is the center of regular pentagon PENTA. What is the image of E for a 1448 rotation about X?

14. The center and vertices divide PENTA into congruent triangles.

15. Divide 3608 by to find the measure of each central angle.

16. Each central angle measures 8.


17. A 1448 rotation is one / two / three times the rotation of the measure in Exercise 16.

18. A 1448 rotation moves each vertex counterclockwise two / three vertices.

19. Circle the image of E for a 1448 rotation about X.

P E N T A

Finding an Angle of Rotation

Got It? Hubcaps of car wheels often have interesting designs that involve rotations. What is the angle of rotation about C that maps M to Q?

20. The hubcap design has spokes that divide the circle into congruent parts.

21. The angle at the center of each part is 3608 4 5 8.

22. As M rotates counterclockwise about C to Q, M touches spokes.

23. As M rotates counterclockwise about C to Q, M rotates through

? 8, or 8.

24. The angle of rotation about C that maps M to Q is 8.

P

E

NT

A

X

4

5

5

72

9

9 40

9

6 40 240

240

6

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Lesson Check

Problem 4

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

x

y

O 2

2

4

2

2

4A’

A

180

233 Lesson 9-3


rotation center of rotation angle of rotation center of a regular polygon

Rate how well you can draw and identify rotation images.


Compare and Contrast Compare rotating a figure about a point to reflecting the figure across a line. How are the transformations alike? How are they different?

28. Rotate nRST 908 about the origin. 29. Reflect nRST across the y-axis.

x

y

O 42

2

4

24

S’

R’ R

T

ST’

x

y

O 42

2

4

24

T’

R’ R

T

SS’

30. Circle the transformation(s) that preserve the size and shape of the preimage. Underline the transformation(s) that preserve the orientation of the preimage.

reflection across a line rotation about a point

31. How are rotating and reflecting a figure alike? How are they different?

__________________________________________________________________________________

Finding a Composition of Rotations

Got It? What are the coordinates of the image of point A(22, 3) for a composition of two 908 rotations about the origin?

25. The composition of two 908 rotations is one 81 8, or 8 rotation.

26. Complete each step to locate point Ar on the diagram at the right.

Step 1 Draw AO.

Step 2 Use a protractor to draw a 1808 angle with the vertex at O and side OA.

Step 3 Use a compass to construct OAr > OA. Graph point Ar .

27. The coordinates of Ar are ( , ). 2 23

90 90 180


The size and shape of the figure are the same. A reflection reverses the orientation.

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Vocabulary

Review

B

B’

D E

FCC’

A

A’

A

A A

Chapter 9 234

Symmetry9-4

1. Circle the center of rotation for the transformation at the right.

A B C

D E F

2. If S is the center of rotation of a figure that contains point Y,

then SY' 5 .

3. Cross out the figure(s) for which point A is NOT the center of rotation.

Vocabulary Builder

symmetry (noun) SIM uh tree

Related Word: symmetrical (adjective)

Math Usage: A figure has symmetry if there is an isometry that maps the figure onto itself. Figures having symmetry are symmetrical.

Use Your Vocabulary

4. Complete each statement with the appropriate form of the word symmetry.

NOUN Some figures have rotational 9.

ADJECTIVE A figure that maps onto itself is 9.


5. A figure that is its own image is symmetrical / symmetry .

6. A line of reflection is a line of symmetrical / symmetry .

7. A butterfly’s wings are symmetrical / symmetry .

SY

symmetry

symmetrical

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Problem 1

Key Concept Types of Symmetry

120

180

235 Lesson 9-4

Identifying Lines of Symmetry

Got It? Draw a rectangle that is not a square. How many lines of symmetry does your rectangle have?

11. Circle the figure that is a rectangle but not a square.

12. Draw a rectangle that is not a square on the grid at the right.

13. Lines of symmetry divide a figure into congruent parts.

14. Draw the line(s) of symmetry on your rectangle.

15. Does your rectangle have a vertical line of symmetry?

Yes / No

16. Does your rectangle have a horizontal line of symmetry?

Yes / No

17. Does your rectangle have a diagonal line of symmetry?

Yes / No

18. A rectangle that is not a square has line(s) of symmetry.

A fi gure has line symmetry or refl ectional symmetry if there is a refl ection for which the fi gure is its own image. Th e line of refl ection is called a line of symmetry. It divides the fi gure into congruent halves.

8. Does the trapezoid at the right have a horizontal line of symmetry? Yes / No

A fi gure has rotational symmetry if there is a rotation of 180° or less for which the fi gure is its own image. Th e angle of rotation for rotational symmetry is the smallest angle needed for the fi gure to rotate onto itself.

9. Is the measure of the angle of rotation for an equilateral triangle 60°? Yes / No

A fi gure with 180° rotational symmetry also has point symmetry. Each segment joining a point and its 180° rotation image passes through the center of rotation.

A square, which has both 90° and 180° rotational symmetry, also has point symmetry.

10. Is the center of rotation of the parallelogram at the right equidistant from all vertices? Yes / No

Drawings may vary, but should have one vertical and one horizontal line of symmetry.

2

2

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Problem 3

Problem 2

P P

Chapter 9 236

Identifying a Rotational Symmetry

Got It? Does the figure at the right have rotational symmetry? If so, what is the angle of rotation?


Each / No side is horizontal or vertical.

20. To keep the same orientation of the sides, the angle of rotation must be

a multiple of 8.

21. Draw the image after a 908 rotation 22. Draw the figure after a 1808 rotationabout point P. Two sides are drawn about point P. Two sides are drawnfor you. for you.

23. Does the figure have rotational symmetry? If so, what is the angle of rotation?

_______________________________________________________________________

Identifying Symmetry in a Three-Dimensional Object

Got It? Does the lampshade have reflectional symmetry in a plane, rotational symmetry about a line, or both?


24. A plane that is parallel to the top of the lampshade and passes through its middle divides the lampshade into two congruent parts.

25. A plane that is perpendicular to the top of the lampshade and passes through its middle divides the lampshade into two congruent parts.

26. The lampshade can be rotated about a horizontal line so that it matches perfectly.

27. The lampshade can be rotated about a vertical line so that it matches perfectly.

28. Does the lampshade have reflectional symmetry in a plane, rotational symmetry about a line, or both? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

90

Yes; 1808

F

F

T

T

Both. Explanations may vary. Sample: The lampshade has

reflectional symmetry in a vertical plane and rotational symmetry

about a vertical line.

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Now Iget it!

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0 2 4 6 8 10

Math Success

Lesson Check

237 Lesson 9-4


symmetry line of symmetry

Rate how well you can identify symmetry in a fi gure.


Error Analysis Your friend thinks that the regular pentagon in the diagram at the right has 10 lines of symmetry. Explain and correct your friend’s error.

29. Place a ✓ in the box if the response is correct. Place an ✗ if it is incorrect.

Each line of symmetry bisects an angle of the pentagon.

Each line of symmetry bisects a side of the pentagon.

Each line of symmetry is parallel to another line of symmetry.

The pentagon has 10 congruent angles.

The pentagon has 10 congruent sides.

The pentagon has 5 congruent angles.

The pentagon has 5 lines of symmetry.

30. Explain your friend’s error.

________________________________________________________________________

________________________________________________________________________

31. Use the regular pentagon below. Draw each line of symmetry in a different color.

✔

✔

✔

✘

✘

✘

Answers may vary. Sample: Your friend counted each

arrow, not each line.

✔

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Vocabulary

Review

Chapter 9 238

9-5 Dilations

Complete each statement with ratio or similar.

1. The 9 of corresponding parts of similar figures is the scale factor.

2. You can use a scale factor to make a larger or smaller copy that is 9 to the original figure.

3. Circle the scale factor that makes an image larger than the preimage.

23 4

3 78 1

10

4. Circle the scale factor that makes an image smaller than the preimage.

52 9

2 14 3

Vocabulary Builder

dilation (noun) dy LAY shun

Definition: A dilation is the widening of an object such as the pupil of an eye or a blood vessel.

Math Usage: A dilation is a transformation that reduces or enlarges a figure so that the image is similar to the preimage.

Related Words: reduction, enlargement, scale factor, center of dilation

Examples: an enlargement of a photograph, a model of the solar system

Use Your Vocabulary


A dilation is an enlargement if the figure decreases / increases in size.

6. Cross out the transformation that does NOT have a center.

reflection rotation dilation

7. Circle the transformations that are isometries.

reflection rotation dilation

ratio

similar

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Problem 1

Key Concept Dilation

C C

P

PQQ

RR8CR n CR

2 2 4K

LM

JJ

K

LM

x

y

3

O

239 Lesson 9-5

Finding a Scale Factor

Got It? J9K9L9M9 is a dilation image of JKLM. The center of dilation is O. Is the dilation an enlargement or a reduction? What is the scale factor of the dilation?


9. The image is larger / smaller than preimage.

10. The dilation is a(n) enlargement / reduction .

11. How can you tell which segments are corresponding sides of JKLM and JrKrLrMr?

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12. Circle the side that corresponds to JK .

JrKr JrMr LrKr

13. Find the length of each side.

JK 5 Ä( 2 )2 1 ( 2 )2 5 Ä

JrKr 5 Ä( 2 )2 1 ( 2 )2 5 Ä

14. Find the scale factor.

JK

JrKr5 5 5

15. The scale factor is .

A dilation with center C and scale factor n, n . 0, is a transformation with these two properties:

• Th e image of C is itself (that is, C r 5 C).

• For any other point R, R ris on CR) and

CR r 5 n ? CR, or n 5 CRrCR .

Th e image of a dilation is similar to its preimage.

8. For a dilation of nPQR with scale factor 2, CR r 5 ? CR.

!

2

Answers may vary. Sample: Corresponding sides have the same

letters as endpoints and are listed in the same order.

6

3 0 0 1

1

4

1

2

10

"10

4000 2

"40

12

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Problem 3

x

y

O 2 3

1

1

1

2

23 P’

G’

G

Z’

P”

Z

P

Problem 2

2y

6 4 4Ox

Z

P

G

Chapter 9 240

Finding a Dilation Image

Got It? What are the images of the vertices of kPZG for a dilation with center (0, 0) and scale factor 12?

16. Complete the problem-solving model below.

17. Use the dilation rule to find the coordinates of the images of the vertices.

P( , ) S P r( , )

Z( , ) S Z r( , )

G( , ) S G r( , )

18. Graph the images of the vertices of nPZG on the coordinate plane. Graph nPrZrGr.

Using a Scale Factor to Find a Length

Got It? The height of a document on your computer screen is 20.4 cm. When you change the zoom setting on your screen from 100% to 25%, the new image of your document is a dilation of the previous image with scale factor 0.25. What is the height of the new image?


The scale factor 0.25 is less than 1, so the dilation is a(n) enlargement / reduction .

20. Image length 5 scale factor ∙ original length, so image height 5 ? ,

or cm.

Know Need PlanCoordinates of vertices:

P(2, 0), Z( , ),

and G( , )

Center of dilation:

( , )

Scale factor:

Coordinates of the images of the vertices

Substitute the coordinates of the vertices into the

dilation rule: (x, y) S

( ? x, ? y)

12

12

1223

2

12

1

1

1

23

23

0.25

5.1

20.4

0

2

0

0

0 0

0

0

22

22 21

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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

Lesson Check

2

6

1

3

A

26

13=n = 4

1 = 4n =

241 Lesson 9-5



dilation center of dilation scale factor of a dilation

enlargement reduction

Rate how well you understand dilation images of fi gures.

Error Analysis The blue figure is a dilation image of the black figure for a dilation with center A.

Two students made errors when asked to find the scale factor. Explain and correct their answers.

A. B.


21. The dilation is an enlargement.

22. The side lengths of the black triangle are 6 and 3.

23. The side lengths of the blue triangle are 2 and 1.

24. The scale factor is between 0 and 1.

25. Explain the error the student made in solution A.

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26. Explain the error the student made in solution B.

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_______________________________________________________________________

27. The correct scale factor is .14

T

T

F

F

Answers may vary. Sample: The student did not use the length 8 of

the black triangle.

Answers may vary. Sample: The student didn’t write the image

length in the numerator.

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Vocabulary

Review

Chapter 9 242

Compositions ofReflections9-6


1. A reflection flips a figure across a line of reflection.

2. A reflection turns a figure about a point.

3. A reflection preimage and image are congruent.

4. The orientation of a figure reverses after a reflection.

5. A line of reflection is either horizontal or vertical.

Vocabulary Builder

composition (noun) kahm puh ZISH un

Other Word Forms: compose (verb), composite (adjective), composite (noun)

Definition: A composition combines parts.

Math Usage: A composition of transformations combines two or more transformations in a given order.

Use Your Vocabulary

Complete each statement with the appropriate word from the list. Use each word only once.

reflections rotation symmetry

6. A composition of reflections has at least one line of 9.

7. You can map any congruent figure onto another using a composition of 9.

8. A composition of rotations is always a 9.

T

T

T

F

F

symmetry

reflections

rotation

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Problem 1

Theorem 9-1 A translation or rotation is a composition of two refl ections.

Theorem 9-2

A composition of refl ections across A composition of refl ections across parallel lines is a translation. two intersecting lines is a rotation.

Theorem 9-1 and Theorem 9-2

P’

P”

PA

, m

B

RR

R

243 Lesson 9-6

Composing Reflections Across Parallel Lines

Got It? Lines < and m are parallel. R is between < and m. What is the image of R reflected first across line < and then across line m? What are the direction and distance of the resulting translation?

9. The diagram shows a dashed line perpendicular to / and m that intersects / at point A, m at point B, and R only at point P. Complete each step to show the composition of the reflections.

Step 1 Refl ect R across line /. Point Prshould correspond to point P.

Step 2 Refl ect the image across line m. Point Ps should correspond to point Pr .

10. Underline the correct word to complete each sentence.

Th e translation is to the right / left along the dashed line.

Th e direction of the translation is parallel / perpendicular to lines / and m.

11. Use the justifications at the right to find the distance PPs of the resulting translation.

PPs 5 1 BPs Segment Addition Postulate

5 1 BPr Definition of reflection across line m

5 1 (BP 1 PA 1 APr) Segment Addition Postulate

5 1 BP 1 2PA Definition of reflection across line /

5 ? BP 1 2PA Simplify.

5 ? (BP 1 PA) Use the Distributive Property.

5 ? Segment Addition Postulate

12. The resulting translation moved R a distance of .

PB

PB

PB

PB

2

2

2 AB

2AB

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Problem 3

Theorem 9–4 Isometry Classification Theorem

Translation

Orientations are the same. Orientations are opposite.

Rotation Reflection Glide Reflection

R RRRRR RR

R

x

y

O2 2 4 646

2

4

2

4

6

8

T

T’

X

X’

E

E’

Chapter 9 244

Finding a Glide Reflection Image

Got It? What is the image of kTEX for a glide reflection where the translation is (x, y)m (x 1 1, y) and the line of reflection is y 5 22?

Use the coordinate plane at the right for Exercises 14–17.

14. Find the vertices of the translation image. Then graph the translation image.

T(25, 2) S (25 1 , ) 5 ( , )

E(21, 3) S (21 1 , ) 5 ( , )

X(22, 1) S (22 1 , ) 5 ( , )

15. In a reflection across a horizontal line,

only the -coordinate changes.

16. Find the vertices of the triangle you graphed in Exercise 14 after reflection across

the line y 5 22.

( , ) S T r( , )

( , ) S Er( , )

( , ) S Xr( , )

17. The image of nTEX for the given glide reflection is the triangle with vertices

T r( , ), Er( , ), and Xr( , ). Graph nT rErXr.

In a plane, one of two congruent fi gures can be mapped onto the other by a composition of at most three refl ections.


If two congruent fi gures in a plane have opposite orientations, an even / odd number of refl ections maps one fi gure onto the other.

Theorem 9–3 Fundamental Theorem of Isometries

Th ere are only four isometries.

1 2 24

21

21

21

21

2

3031

1 1 1

y

24 2 24 26

26

0 3 0 27

27

1 25

25024

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Lesson Check

Now Iget it!

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0 2 4 6 8 10

Math Success

PP

PP

Problem 4

m

n

DF

E

opposite

same

translation

rotation

reflection

glide reflection

245 Lesson 9-6

Error Analysis You reflect kDEF first across line m and then across line n. Your friend says you can get the same result by reflecting kDEF first across line n and then across line m. Explain your friend’s error.

21. Place a ✓ in the box if the response is correct. Place an ✗ if it is incorrect.

Lines m and n are perpendicular.

A clockwise or counterclockwise rotation has the same image.

22. Explain your friend’s error.

_______________________________________________________________________


Classifying Isometries

Got It? Each figure is an isometry image of the figure at the right. Are the orientations of the preimage and image the same or opposite? What type of isometry maps the preimage to the image?

A. B. C.

Choose the correct words from the list to complete each sentence.

18. Image A has the 9 orientation and is a 9.

19. Image B has the 9 orientation and is a 9.

20. Image C has the 9 orientation and is a 9.


composition of refl ections glide refl ection isometry

Rate how well you can fi nd compositions of refl ections.

✗

✗

The resulting rotation is not in the same direction.


same rotation

same translation

opposite glide reflection

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Vocabulary

Review

Tessellations

Chapter 9 246

9-7


A square has two / four lines of symmetry.

2. Circle the figure that has exactly two lines of symmetry.

circle equilateral triangle isosceles triangle rectangle square

Vocabulary Builder

tessellation (noun) tes uh LAY shun

Related Words: tessellate (verb), tiling (noun)

Definition: A tessellation is a repeated pattern of figures that completely covers a plane, without gaps or overlaps.

Main Idea: You can identify the transformations and symmetries in tessellations.

Example: Squares make a tesselation because laying them side by side completely covers the plane without gaps or overlaps.

Non-Example: Circles cannot make a tesselation because they leave gaps when placed so they touch but do not overlap.

Use Your Vocabulary


3. A tessellation of two figures may overlap.

4. You can make a tessellation with translations.

5. You cannot make a tessellation with reflections.

6. You can use any two figures to make a tessellation.

7. A tiled floor is an example of a tessellation.

tessellation

T

T

F

F

F

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d.Problem 1

Problem 2

247 Lesson 9-7

Describing Tessellations

Got It? What is the repeating figure in the tessellation? What transformation does the tessellation use?


The repeating figure in the tessellation is a lizard / bird / turtle .

9. Describe the orientation of repeating figures in the tessellation.

_______________________________________________________________________

_______________________________________________________________________

10. Circle the transformation used in the tessellation.

glide reflection reflection rotation translation

Determining Whether a Figure Tessellates

Got It? Does a regular hexagon tessellate? Explain.

11. Place a ✓ in the box if the statement is correct. Place an ✗ if it is incorrect.

If the measure of one angle of a regular polygon is a factor of 3608, the polygon tessellates.

If the sum of the measures of n angles of a regular polygon is less than 3608, there are gaps when n copies of the polygon are placed around a vertex.

If the sum of the measures of n angles of a regular polygon is greater than 3608, there are gaps when n copies of the polygon are placed around a vertex.

12. The sum of the measures of the angles of a regular hexagon is 8.

13. The measure of each angle of a regular hexagon is 8.

14. Does a regular hexagon tessellate? Explain.

_______________________________________________________________________

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_______________________________________________________________________


Repeating figures all look to the right, so they have the

same orientation.

Yes. Explanations may vary. Sample: The measure of each angle is 1208

so 3 copies of the hexagon will exactly fill the 3608 around any vertex.

✓

✗

✓

120

720

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Problem 3

Tessellation A Tessellation B

Chapter 9 248

Identifying Symmetries in a Tessellation

Got It? What types of symmetry does the tessellation at the right have?

Draw a line from the type of symmetry in Column A to its description in Column B.

Column A Column B

15. glide reflectional symmetry A figure maps onto itself after a turn.

16. reflectional symmetry A figure is its own image after moving a given distance.

17. rotational symmetry A line of symmetry divides the figure into two congruent halves.

18. translational symmetry A figure maps onto itself after a translation and a reflection.

Use the diagrams below for Exercises 19–23.

19. Circle the type of symmetry shown by the red point and arc in Tessellation A.

glide reflectional reflectional rotational translational

20. Circle the type of symmetry shown by the blue line in Tessellation A.


21. Circle the type of symmetry shown by the red arrow in Tessellation B.


22. Circle the type of symmetry shown by the blue arrow and dashed line in tessellation B.


23. What types of symmetry does the tessellation have?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

The tessellation has glide reflectional symmetry,

reflectional symmetry, rotational symmetry,

and translational symmetry.

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Now Iget it!

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0 2 4 6 8 10

Math Success

Think Write

I know that all angles of a regular polygon

have the same measure.

a 135

a 8

180(8 2)

I can use the Polygon Angle-Sum Theorem

to find a.

Let a the measure of one angle.

8I know that a regular octagon has

sides. I can substitute for n.

Now I can simplify.

3

Finally, I should find the total measure

of angles.

n

na

180( 2)

3 135 405?

8

249 Lesson 9-7

Reasoning If you arrange three regular octagons so that they meet at one vertex, will they leave a gap or will they overlap? Explain.

24. Complete the reasoning model below.

25. If you arrange three regular octagons so that they meet at one vertex, will they leave a gap or will they overlap? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________



tessellation tiling

Rate how well you can identify figures that tessellate.

They will overlap. Explanations may vary. Sample: The

sum of the measures of the three angles is 405, which is greater than

360.

9-1 translations - trimble county schools · 9-1 translations review 1. ... cross out the word that...

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