4.1 Practice A want to plot the collinear points AAxyA()() ()−2, 3 , , , and 3, 7′′′on the...

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111

4.1 Practice A

Name _________________________________________________________ Date __________

1. Name the vector and write its component form.

2. The vertices of ABC are ( ) ( ) ( )2, 3 , 1, 2 , and 0, 1 .A B C− Translate ABC using

the vector 1, 4 .− Graph ABC and its image.

3. Find the component form of the vector that translates ( ) ( )3, 2 to 1, 4 .A A′− −

4. Write a rule for the translation of RST to .R S T′ ′ ′

In Exercises 5 and 6, use the translation ( ) ( ), 1, 3x y x y→ + − to find the image

of the given point.

5. ( )5, 9Q 6. ( )3, 8M − −

In Exercises 7 and 8, graph CDE with vertices ( ) ( ) ( )C D E1, 3 , 0, 2 , and 1, 1− − and

its image after the given translation or composition.

7. Translation: ( ) ( ), 3, 1x y x y→ − + 8. Translation: ( ) ( ), 10, 8x y x y→ + −

Translation: ( ) ( ), 7, 15x y x y→ − +

9. You want to plot the collinear points ( ) ( ) ( )2, 3 , , , and 3, 7A A x y A′ ′′− on the same coordinate plane. Do you have enough information to find the values of x and y? Explain your reasoning.

10. You are using the map shown to navigate through the city. You decide to walk to the Post Office from your current location at the Community Center. Describe the translation that you will follow. If each grid on the map is 0.05 mile, how far will you travel?

J

K

x

y4

−4

6RS

R′S′

T

T′

CommunityCenter

PostOffice

00

1

2

3

4

1 2 3

Geometry Copyright © Big Ideas Learning, LLC Resources by Chapter All rights reserved. 112

4.1 Practice B

Name _________________________________________________________ Date _________

1. The vertices of FGH are ( ) ( ) ( )2, 6 , 3, 0 , and 1, 4 .F G H− − − Translate FGHusing the vector 2, 7 .− Graph FGH and its image.

2. Find the component form of the vector that translates ( ) ( )4, 8 to 7, 9 .A A′− −

3. Write a rule for the translation of ABC to .A B C′ ′ ′

In Exercises 4 and 5, use the translation ( ) ( ), 4, 3x y x y→ − + to find the

image of the given point.

4. ( )2, 4G − 5. ( )10, 5H −

6. Graph JKL with vertices ( ) ( ) ( )2, 8 , 1, 3 , and 5, 4J K L− − and its image after the composition.

Translation: ( ) ( ), 6, 1x y x y→ + −

Translation: ( ) ( ), 1, 7x y x y→ − −

7. Is the transformation given by ( ) ( ), 2 2, 1x y x y→ + + a translation? Explain your reasoning.

8. A popular kid’s game has 15 tiles and 1 open space. The goal of the game is to rearrange the tiles to put them in order (from least to greatest, starting at the upper left-hand corner and going across each row). Use the figure to write the transformation(s) that describe the path of where the 8 tile is currently, and where it must be by the end of the game. Can this same translation be used to describe the path of all the tiles?

9. Graph any triangle and translate it in any direction. Draw translation vectors for each vertex of the triangle. Is there a geometric relationship between all the translation vectors? Explain why this makes sense in terms of the slope of the line.

10. Point ( )4, 2P − undergoes a translation given by ( ) ( ), 3, ,x y x x a→ + − followed by

another translation ( ) ( ), , 7x y x b x→ − + to produce the image of ( )5, 8 .P′′ − Find the values of a and b and point .P′

x

y

−4

8−4

B

AB′

C

A′

C′

8 2 3 75 6 4 141 9 1311 15 10 12


4.2 Practice A

Name _________________________________________________________ Date _________

In Exercises 1–3, graph ABC and its image after a reflection in the given line.

1. ( ) ( ) ( )0, 2 , 1, 3 , 2, 4 ; -axisA B C x−

2. ( ) ( ) ( )2, 4 , 6, 2 , 3, 5 ; -axisA B C y− − −

3. ( ) ( ) ( )4, 1 , 3, 8 , 1, 1 ; 2A B C y− − = −

In Exercises 4 and 5, graph the polygon and its image after a reflection in the given line.

4. y x= − 5. y x=

In Exercises 6 and 7, graph JKL with vertices ( ) ( ) ( )J K L2, 3 , 2, 1 , and 1, 5− − and

its image after the glide reflection.

6. Translation: ( ) ( ), 1, x y x y→ − 7. Translation: ( ) ( ), 2, 3x y x y→ + − Reflection: in the x-axis Reflection: in the line 2x = −

In Exercises 8 and 9, determine the number of lines of symmetry for the figure.

8. 9.

10. Find point W on the y-axis so that VW XW+ is a minimum given ( )2, 3V and

( )2, 1 .X − −

11. A line 3 5y x= − is reflected in x a= so that the image is given by 1 3 .y x= − What is the value of a?

12. Your friend claims that it is not possible to have a glide reflection if you have two translations followed by one reflection. Is your friend correct? Explain your reasoning.

x

y

4

−8

−4

8−8

P

S

Q

R

RS

Q

x

y8

84−4−8 P


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4.2 Practice B

Name _________________________________________________________ Date __________

In Exercises 1 and 2, graph CDE and its image after a reflection in the given line.

1. ( ) ( ) ( )3, 4 , 2, 1 , 0, 5 ; -axisC D E y− − 2. ( ) ( ) ( )1, 6 , 12, 2 , 7, 8 ; 8C D E x− =

In Exercises 3 and 4, graph the polygon and its image after a reflection in the given line.

3. x-axis 4. 1y = −

In Exercises 5 and 6, graph ABC with vertices ( ) ( ) ( )A B C1, 4 , 2, 1 , and 4, 3− − and

its image after the glide reflection.

5. Translation: ( ) ( ), 2, 1x y x y→ + − 6. Translation: ( ) ( ), 3, 1x y x y→ − + Reflection: in the line y x= Reflection: in the line y x= −

7. Determine the number of lines of symmetry 8. Find point P on the x-axis so that for the figure. AP BP+ is a minimum.

9. Is it possible to perform two reflections of an object so that the final image is identical to the original image? If so, give an example. If not, explain your reasoning.

10. A triangle undergoes a glide reflection. Is it possible for the sides of the triangle to change length during this process? Explain your reasoning.

11. Your friend claims that it is not possible to have a glide reflection if you have one translation followed by two reflections. Is your friend correct? Explain your reasoning.

x

y

4

2

6

42−2

L

KM

N

x

y8

4

−4

−4

N

ML

K

x

y4

2

−2

42−2

B A


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4.3 Practice A

Name _________________________________________________________ Date __________

1. Trace the polygon and point P. Then draw a 60° rotation of the polygon about point P.

2. Graph the polygon and its image after a 270° rotation about the origin.

In Exercises 3 and 4, graph RST with vertices ( ) ( ) ( )R S T2, 3 , 2, 1 , and 1, 5− −

and its image after the composition.

3. Translation: ( ) ( ), 2, 1x y x y→ − − 4. Reflection: in the line x y=

Rotation: 90° about the origin Rotation: 180° about the origin

In Exercises 5 and 6, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself.

5. 6.

7. Draw AB with points ( ) ( )2, 0 and 0, 2 .A B Rotate the segment 90° counterclockwise about point A. Then rotate the two segments 180° about the origin. What geometric figure did you create using the original segment and its images?

8. List the uppercase letters of the alphabet that have rotational symmetry, and state the angle of the symmetry.

A

B

C

DP

x

y4

−2

4−2

J

K

L

M


4.3 Practice B

Name _________________________________________________________ Date _________

1. Graph the polygon and its image after a 90° rotation about the origin.

In Exercises 2 and 3, graph CDE with vertices ( ) ( ) ( )C D E1, 3 , 4, 2 , and 5, 1− − − −

and its image after the composition.

2. Rotation: 180° about the origin 3. Reflection: in the line x y=

Translation: ( ) ( ), 3, 1x y x y→ + + Rotation: 270° about the origin

In Exercises 4 and 5, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself.

4. 5.

6. Is it possible to have an object that does not have 360° of rotational symmetry? Explain your reasoning.

7. A figure that is rotated 60° is mapped back onto itself. Does the figure have rotational symmetry? Explain. How many times can you rotate the figure before it is back where it started?

8. Your friend claims that he can do a series of translations on any geometric object and get the same result as a rotation. Is your friend correct?

9. Your friend claims that she can do a series of reflections on any geometric object and get the same result as a rotation. Is your friend correct?

10. List the digits from 0–9 that have rotational symmetry, and state the angle of the symmetry.

x

y

4

−4

84−4

T

V

W

U


4.4 Practice A

Name _________________________________________________________ Date _________

In Exercises 1 and 2, identify any congruent figures in the coordinate plane. Explain.

1. 2.

In Exercises 3 and 4, describe a congruence transformation that maps ABC to A B C .′ ′ ′

3. 4.

In Exercises 5 and 6, determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning.

5. ( ) ( ) ( ) ( ) ( ) ( )5, 2 , 2, 2 , 2, 7 and 4, 5 , 1, 5 , 1, 0A B C S T U− − − − −

6. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )6, 2 , 10, 2 , 10, 8 , 6, 8 and 4, 8 , 4, 10 , 8, 10 , 8, 8E F G H W X Y Z− − − −

7. In the figure, ,a b CDE is reflected in line a, and C D E′ ′ ′ is reflected

in line b. List three pairs of segments that are parallel to each other. Then

determine whether any segments are congruent to .EE′′

In Exercises 8 and 9, find the measure of the acute or right angle formed by intersecting lines so that P can be mapped to P ′′ using two reflections.

8. A rotation of 28° maps to .P P′′

9. The rotation ( ) ( ), , maps to .x y y x P P′′→ −

x

y

2

−4

−2

42−2−4−6

7

2

4

6

13

5

8

x

y

2

4

6

−4

4 6−2−4−6−8

7

9

2

4

6

1

10

3

5

8

−2

−6

x

y4

−8

8−8

B′A

B

A′

CC′ x

y

4−8

A

B

A′

C

B′ C′

−6

C

C′

C″

E″D″

E′D′

D E

a

b


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4.5 Practice A

Name _________________________________________________________ Date __________

In Exercises 1 and 2, find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement.

1. 2.

In Exercises 3–5, copy the diagram. Then use a compass and straightedge to construct a dilation of quadrilateral ABCD with the given center and scale factor k.

3. Center B, 3k =

4. Center P, 12k =

5. Center C, 75%k =

In Exercises 6 and 7, graph the polygon and its image after a dilation with a scale factor k.

6. ( ) ( ) ( ) ( )1, 2 , 2, 2 , 4, 2 , 1, 3 ; 2P Q R S k− − − =

7. ( ) ( ) ( ) ( )4, 4 , 2, 6 , 1, 1 , 2, 4 ; 75%A B C D k− − − − − = −

8. A standard piece of paper is 8.5 inches by 11 inches. A piece of legal-size paper is 8.5 inches by 14 inches. By what scale factor k would you need to dilate the standard paper so that you could fit two pages on a single piece of legal paper?

9. The old film-style cameras created photos that were best printed at 3.5 inches by 5 inches. Today’s new digital cameras create photos that are best printed at 4 inches by 6 inches. Neither size picture will scale perfectly to fit in an 11-inch by 14-inch frame. Which type of camera will you minimize the loss of the edges of your picture?

10. Your friend claims that if you dilate a rectangle by a certain scale factor, then the area of the object also increases or decreases by the same amount. Is your friend correct? Explain your reasoning.

11. Would it make sense to state “A dilation has a scale factor of 1?” Explain your reasoning.

x

y

A B

CD

P

14 in.

8.5 in.

9C

P

P′

27

4

C

P

P′10


4.5 Practice B

Name _________________________________________________________ Date _________

In Exercises 1 and 2, find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement.

1. 2.

In Exercises 3 and 4, copy the diagram. Then use a compass and straightedge to construct a dilation with the given center and scale factor k.

3. Center B, 2k =

4. Center P, 75%k =

In Exercises 5 and 6, graph the polygon and its image after a dilation with a scale factor k.

5. ( ) ( ) ( ) ( )3, 4 , 2, 1 , 3, 2 , 5, 4 ; 50%J K L M k− − − − =

6. ( ) ( ) ( ) ( ) ( )1, 1 , 1, 0 , 4, 2 , 3, 4 , 0, 3 ; 3V W X Y Z k− − − = −

7. You look up at the sky at night and see the moon. It looks like it is about 2 millimeters across. If you then look at the moon through a telescope that has a magnification of 40 times, how big will it look to you through the telescope?

8. What would it mean for an object to be dilated with a scale factor of 0?k =

9. Your friend claims that if you dilate a rectangle by a certain scale factor, then the perimeter of the object also increases or decreases by the same factor. Is your friend correct? Explain your reasoning.

10. The image shows an object that has been dilated with an unknown scale factor. Use the given measures to determine the scale factor and solve for the value of x.

C

2

3x + 2

2x

16

A

A′

P B

C

D

A

9P

P′

C1.5

12P

P′

C8


4.6 Practice A

Name _________________________________________________________ Date _________

In Exercises 1 and 2, graph PQR with vertices ( ) ( ) ( )P Q R1, 5 , 4, 3 , and 2, 1− − −

and its image after the similarity transformation.

1. Rotation: 180° about the origin 2. Dilation: ( ) ( )1 12 2, , x y x y→

Dilation: ( ) ( ), 2 , 2x y x y→ Reflection: in the x-axis

3. Describe a similarity transformation that maps the black preimage onto the dashed image.

In Exercises 4 and 5, determine whether the polygons with the given vertices are similar. Use transformations to explain your reasoning.

4. ( ) ( ) ( )( ) ( ) ( )

2, 5 , 2, 2 , 1, 2 and

3, 3 , 3, 1 , 2, 1

A B C

D E F

− − − 5. ( ) ( ) ( ) ( )( ) ( ) ( ) ( )

5, 3 , 3, 1 , 3, 5 , 5, 5 and

3, 3 , 4, 3 , 4, 2 , 3, 1

J K L M

T U V W

− − − − − − − −

6. Prove that the figures are similar.

Given Equilateral GHI with side length a, equilateral PQR with side length b

Prove GHI is similar to .PQR

7. Your friend claims you can use a similarity transformation to turn a square into a rectangle. Is your friend correct? Explain your answer.

8. Is the composition of a dilation and a translation commutative? In other words, do you obtain the same image regardless of the order in which the transformations are performed? Justify your answer.

9. The image shown is known as a Sierpinski triangle. It is a common mathematical construct in the area of fractals. What can you say about the similarity transformations used to create the white triangles in this image?

x

y

4−8

YY′

X′ W′

−8

XW

Z

Z′

G

HI

Q

R

Pa

b

4.1 Practice A want to plot the collinear points AAxyA()() ()−2, 3 , , , and 3, 7′′′on the...

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