# Big Ideas Math Geometry Answers Chapter 4 Transformations

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## Big Ideas Math Book Geometry Answer Key Chapter 4 Transformations

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### Transformations Maintaining Mathematical Proficiency

Tell whether the red figure is a translation, reflection, rotation, or dilation of the blue figure.

Question 1.

Question 2.

Question 3.

Question 4.

Tell whether the two figures are similar. Explain your reasoning.

Question 5.

Question 6.

Question 7.

### Transformations Mathematical Practices

Monitoring Progress

Use dynamic geometry software to draw the polygon with the given vertices. Use the
software to find the side lengths and angle measures of the polygon. Round your answers to time nearest hundredth.

Question 1.
A(0, 2), B(3, – 1), C(4, 3)

Question 2.
A(- 2, 1), B(- 2, – 1), C(3, 2)

Question 3.
A(1, 1), B(- 3, 1), C(- 3, – 2), D(1, – 2)

Question 4.
A(1, 1) B(- 3, 1), C(- 2, – 2), D(2, – 2)

Question 5.
A(- 3, 0), B(0, 3), C(3, 0), D(0, – 3)

Question 6.
A(0, 0), B(4, 0), C(1, 1), D(0, 3)

### 4.1 Translations

Exploration 1

Translating a Triangle in a Coordinate Plane

Work with a partner.
a. Use dynamic geometry software to draw any triangle and label it ∆ABC.

b. Copy the triangle and translate (or slide) it to form a new figure, called an image, ∆A’B’C’ (read as triangle A prime, B prime. C prime”).
USING TOOLS STRATEGICALLY
To be proficient in math, you need to use appropriate tools strategically, including dynamic geometry software.

c. What is the relationship between the coordinates of the vertices of ∆ABC and
those of ∆A’B’C’?

d. What do you observe about the side lengths and angle measures of the two triangles?

Exploration 2

Translating a Triangle in a Coordinate Plane

Work with a partner.
a. The point (x, y) is translated a units horizontally and b units vertically. Write a rule to determine the coordinates of the image of (x, y).

b. Use the rule you wrote in part (a) to translate ∆ABC 4 units left and 3 units down. What are the coordinates of the vertices of the image. ∆A’B’C’?

c. Draw ∆A’B’C.’ Are its side lengths the same as those of ∆ABC? Justify your answer.

Exploration 3

Comparing Angles of Translations

Work with a partner.

a. In Exploration 2, is ∆ABC a righL triangle? Justify your answer.

b. In Exploration 2, is ∆A’B’C’ a right triangle? Justify your answer.

c. Do you think translations always preserve angle measures? Explain your reasoning.

Question 4.
How can you translate a figure in a coordinate plane?

Question 5.
En Exploration 2. translate ∆A’B’C’ 3 units right and 4 units up. What are the coordinates of the vertices of the image, ∆A”B”C”? How are these coordinates
related to the coordinates of the vertices of the original triangle. ∆ABC?

### Lesson 4.1 Translations

Monitoring Progress

Question 1.
Name the vector and write its component form.

Question 2.
The vertices of ∆LMN are L(2, 2), M(5, 3), and N(9, 1). Translate ∆LMN using the vector (- 2, 6).

Question 3.
In Example 3. write a rule to translate ∆A’B’C’ back to ∆ABC.

Question 4.
Graph ∆RST with vertices R(2, 2), S(5, 2), and T(3, 5) and its image alter the translation (x, y) → (x + 1, y + 2).

Question 5.
Graph $$\overline{T U}$$ with endpoints T(1, 2) and U(4, 6) and its image after the composition.
Translation: (x, y) → (x – 2, y – 3)
Translation: (x, y) → (x – 4, y + 5)

Question 6.
Graph $$\overline{V W}$$ with endpoints V(- 6, – 4) and W(- 3, 1) and its image after the composition.
Translation: (x, y) → (x + 3, y + 1)
Translation: (x, y) → (x – 6, y – 4)

Question 7.
In Example 6, you move the gray square 2 units right and 3 units up. Then you
move the gray square 1 unit left and 1 unit down. Rewrite the composition as a single transformation.

### Exercise 4.1 Translations

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Name the preimage and image of the transformation ∆ABC – ∆A’B’C’.

Question 2.
COMPLETE THE SENTENCE
A _______ moves every point of a figure the same distance in the same direction.

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, name the vector and write its component form.

Question 3.

Question 4.

In Exercises 5 – 8, the vertices of ∆DEF are D(2, 5), E(6, 3), and F(4, 0). Translate ∆DEF using the given vector. Graph ∆DEF and its image.

Question 5.
(6, 0)

Question 6.
(5, – 1)

Question 7.
(- 3, – 7)

Question 8.
(- 2, – 4)

In Exercises 9 and 10, find the component form of the vector that translates P(- 3, 6) to P’.

Question 9.
P'(0, 1)

Question 10.
P'(- 4, 8)

In Exercises 11 and 12, write a rule for the translation of ∆LMN to ∆L’M’W’.

Question 11.

Question 12.

In Exercises 13 – 16, use the translation.
(x, y) → (x – 8,y + 4)

Question 13.
What is the image of A(2, 6)?

Question 14.
What is the image of B(- 1, 5)?

Question 15.
What is the preimage of C'(- 3, – 10)?

Question 16.
What is the preimage of D'(4, – 3)?

In Exercises 17 – 20, graph ∆PQR with vertices P (-2, 3) Q(1, 2), and R(3, – 1) and its image after the translation.
Question 17.
(x, y) → (x + 4, y + 6)

Question 18.
(x, y) → (x + 9, y – 2)

Question 19.
(x, y) → (x – 2, y – 5)

Question 20.
(x, y) → (x – 1, y + 3)

In Exercises 21 and 22. graph ∆XYZ with vertices X(2, 4), Y(6, 0). and Z(7, 2) and its image after the composition.

Question 21.
Translation: (x, y) → (x + 12, y + 4)
Translation: (x, y) → (x – 5, y – 9)

Question 22.
Translation: (x, y) → (x – 6, y)
Translation: (x, y) → (x + 2, y + 7)

In Exercises 23 and 24, describe the composition of translations.

Question 23.

Question 24.

Question 25.
ERROR ANALYSIS
Describe and correct the error in graphing the image of quadrilateral EFGH after the translation (x, y) → (x – 1, y – 2).

Question 26.
MODELING WITH MATHEMATICS
In chess, the knight (the piece shaped like a horse) moves in an L pattern. The hoard shows two consecutive moves of a black knight during a game. Write a composition of translations for the moves. Then rewrite the composition as a single translation that moves the knight from its original position to its ending position.

Question 27.
PROBLEM SOLVING
You are studying an amoeba through a microscope. Suppose the amoeba moves on a grid-indexed microscope slide in a straight line from square B3 to square G7.

a. Describe the translation.
b. The side length of each grid square is 2 millimeters. How far does the amoeba travel?
c. The amoeba moves from square B3 to square G7 in 24.5 seconds. What is its speed in millimeters per second?

Question 28.
MATHEMATICAL CONNECTIONS
Translation A maps (x, y) to (x + n, y + t). Translation B maps (x, y) to (x + s, y + m).
a. Translate a point using Translation A, followed by Translation B. Write an algebraic rule for the final image of the point after this composition.

b. Translate a point using Translation B, followed by Translation A. Write an algebraic rule for the final image of the point after this composition.

c. Compare the rules you wrote for parts (a) and (b) Does it matter which translation you do first? Explain your reasoning.

MATHEMATICAL CONNECTIONS
In Exercises 29 and 30, a translation maps the blue figure to the red figure. Find the value of each variable.
Question 29.

Question 30.

Question 31.
USING STRUCTURE
Quadrilateral DEFG has vertices D(- 1, 2), E(- 2, 0), F(- 1, – 1), and G( 1, 3). A translation maps quadrilateral DEFG to quadrilateral D’E’F’G’. The image of D is D'(- 2, – 2). What are the coordinates of E’, F’, and G’?

Question 32.
HOW DO YOU SEE IT?
Which two figures represent a translation? Describe the translation.

Question 33.
REASONING
The translation (x, y) → (x + m, y + n) maps $$\overline{P Q}$$ to $$\overline{P’ Q’}$$. Write a rule for the translation of $$\overline{P’ Q’}$$ to $$\overline{P Q}$$. Explain your reasoning.

Question 34.
DRAWING CONCLUSIONS
The vertices of a rectangle are Q(2, – 3), R(2, 4), S(5, 4), and T(5, – 3),
a. Translate rectangle QRST 3 units left and 3 units down to produce rectangle Q’R’S’T’. Find the area of rectangle QRST and the area of rectangle Q’R’S’T’.

b. Compare the areas. Make a conjecture about the areas of a preimage and its image after a translation.

Question 35.
PROVING A THEOREM
Prove the Composition Theorem (Theorem 4.1).

Question 36.
PROVING A THEOREM
Use properties of translations to prove each theorem.
a. Corresponding Angles Theorem (Theorem 3. 1)

b. Corresponding Angles Converse (Theorem 3.5)

Question 37.
WRITING
Explain how to use translations to draw a rectangular prism.

Question 38.
MATHEMATICAL CONNECTIONS
The vector PQ = (4, 1) describes the translation of A(- 1, w) Onto A'(2x + 1, 4) and B(8y – 1, 1) Onto B'(3, 3z). Find the values of w, x, y, and z.

Question 39.
MAKING AN ARGUMENT
A translation maps $$\overline{G H}$$ to $$\overline{G’ H’}$$. Your friend claims that if you draw segments connecting G to G’ and H to H’, then the resulting quadrilateral is a parallelogram. Is your friend correct? Explain your reasoning.

Question 40.
THOUGHT PROVOKING
You are a graphic designer for a company that manufactures floor tiles. Design a floor tile in a coordinate plane. Then use translations to show how the tiles cover an entire floor. Describe the translations that map the original tile to four other tiles.

Question 41.
REASONING
The vertices of ∆ABC are A(2, 2), B(4, 2), and C(3, 4). Graph the image of ∆ABC after the transformation (x, y) → (x + y, y). Is this transformation a translation? Explain your reasoning.

Question 42.
PROOF
$$\overline{M N}$$ is perpendicular to line l. $$\overline{M’ N’}$$ is the translation of $$\overline{M N}$$ 2 units to the left. Prove that $$\overline{M’ N’}$$ is perpendicular to l.

Maintaining Mathematical Proficiency

Tell whether the figure can be folded in half so that one side matches the other.

Question 43.

Question 44.

Question 45.

Question 46.

Simplify the expression.

Question 47.
– (- x)

Question 48.
– (x + 3)

Question 49.
x – (12 – 5x)

Question 50.
x – (- 2x + 4)

### 4.2 Reflections

Exploration

Reflecting a Triangle Using a Reflective Device

Work with a partner:
Use a straightedge to draw any triangle on paper. Label if ∆ABC.

a. Use the straightedge to draw a line that does not pass through the triangle. Label it m.

b. Place a reflective device on line in.

c. Use the reflective device to plot the images of the vertices of ∆ABc. Label the images of vertices A, B. and C as A’, B’, and C’, respectively.

d. Use a straightedge to draw ∆A’B’C by connecting the vertices.

Exploration 2

Reflecting a Triangle in a Coordinate Plane

Work with a partner: Use dynamic geometry software to draw any triangle and label it ∆ABC.

a. Reflect ∆ABC in the y-axis to form ∆A’B’C’.

b. What is the relationship between the coordinates of the vertices of ∆ABC and
those of ∆A’B’C’?
LOOKING FOR STRUCTURE
To be proficient in math, you need to look closely to discern a pattern or structure.

c. What do you observe about the side lengths and angle measures of the two triangles?

d. Reflect ∆ABC in the x-axis to form ∆A’B’C’. Then repeal parts (b) and (c).

Question 3.
How can you reflect a figure in a coordinate plane?

### Lesson 4.2 Reflections

Monitoring progress

Graph ∆ABC from Example 1 and its image after a reflection in the given line.
Question 1.
x = 4

Question 2.
x = – 3

Question 3.
y = 2

Question 4.
y = – 1

The vertices of ∆JKL are J(1, 3), K(4, 4), and L(3, 1).

Question 5.
Graph ∆JKL and its image after a reflection in the x-axis.

Question 6.
Graph ∆JKL and its image after a reflection in the y-axis.

Question 7.
Graph ∆JKL and its image after a reflection in the line y = x.

Question 8.
Graph ∆JKL and its image aIter a reflection in the line y = – x.

Question 9.
In Example 3. verify that $$\overline{F F’}$$ is perpendicular to y = – x.

Question 10.
WHAT IF?
In Example 4, ∆ABC is translated 4 units down and then reflected in the y-axis. Graph ∆ABC and its image after the glide reflection.

Question 11.
In Example 4. describe a glide reflection from ∆A”B”C” to ∆ABC.

Determine the number of lines of symmetry for the figure.

Question 12.

Question 13.

Question 14.

Question 15.
Draw a hexagon with no lines of symmetry.

Question 16.
Look back at Example 6. Answer the question by Using a reflection of point A instead of point B.

### Exercise 4.2 Reflections

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
A glide reflection is a combination of which two transformations?

Question 2.
WHICH ONE DOESN’T BELONG?
Which transformation does not belong with the other three? Explain your reasoning.

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, determine whether the coordinate plane shows a reflection in the x-axis, y-axis, or neither.

Question 3.

Question 4.

Question 5.

Question 6.

In Exercises 7 – 12, graph ∆JKL and its image after a
reflection in the given line.
Question 7.
J(2, – 4), K(3, 7), L(6, – 1); x-axis

Question 8.
J(5, 3), K(1, – 2), L(- 3, 4); y-axis

Question 9.
J(2, – 1), K(4, – 5), L(3, 1); x = – 1

Question 10.
J(1, – 1), K(3, 0), L(0, – 4); x = 2

Question 11.
J(2, 4), K(- 4, – 2), L(- 1, 0); y = 1

Question 12.
J(3, – 5), K(4, – 1), L(0, – 3); y = – 3

In Exercises 13-16, graph the polygon and its image after a reflection in the given line.

Question 13.
y = x

Question 14.
y = x

Question 15.
y = -x

Question 16.
y = -x

In Exercises 17-20. graph ∆RST with vertices R(4, 1), s(7, 3), and T(6, 4) and its image after the glide reflection.

Question 17.
Translation: (x, y) → (x, y – 1)
Reflection: in the y-axis

Question 18.
Translation: (x, y) → (x – 3,y)
Reflection: in the line y = – 1

Question 19.
Translation: (x, y) → (x, y + 4)
Reflection: in the line x = 3

Question 20.
Translation: (x, y) → (x + 2, y + 2)
Reflection: in the line y = x

In Exercises 21 – 24, determine the number of lines of symmetry for the figure.

Question 21.

Question 22.

Question 23.

Question 24.

Question 25.
USING STRUCTURE
Identify the line symmetry (if any) of each word.
a. LOOK
b. MOM
c. OX

Question 26.
ERROR ANALYSIS
Describe and correct the error in describing the transformation.

Question 27.
MODELING WITH MATHEMATICS
You park at some point K on line n. You deliver a pizza to House H, go back to your car. and deliver a pizza to House J. Assuming that you can cut across both lawns, how can you determine the parking location K that minimizes the distance HK + KJ?

Question 28.
ATTENDING TO PRECISION
Use the numbers and symbols to create the glide reflection resulting in the image shown.

In Exercises 29 – 32, find point C on the x-axis so AC + BC is a minimum.

Question 29.
A(1, 4), B(6, 1)

Question 30.
A(4, – 5), B(12, 3)

Question 31.
A(- 8, 4), B(- 1, 3)

Question 32.
A(- 1, 7), B(5, – 4)

Question 33.
MATHEMATICAL CONNECTIONS
The line y = 3x + 2 is reflected in the line y = – 1. What is the equation of the image?

Question 34.
HOW DO YOU SEE IT?
Use Figure A.

a. Which figure is a reflection of Figure A in the line x = a? Explain.

b. Which figure is a reflection of Figure A in the line y = b? Explain.

c. Which figure is a reflection of Figure A in the line y = x? Explain.

d. Is there a figure that represents a glide reflection? Explain your reasoning.

Question 35.
CONSTRUCTION
Follow these steps to construct a reflection of △ ABC in line m. Use a compass and straightedge.

Step 1 Draw △ABC and line m.
Step 2 Use one compass setting to find two points that are equidistant from A on line m. Use the same compass setting to find a point on the other side of m that is the same distance from these two points. Label that point as A’.
Step 3 Repeat Step 2 to find points B’ and C’.
Draw △A’B’C.

Question 36.
USING TOOLS
Use a reflective device to verify your construction in Exercise 35.

Question 37.
MATHEMATICAL CONNECTIONS
Reflect △MNQ in the line y = -2x.

Question 38.
THOUGHT PROVOKING
Is the composition of a translation and a reflection commutative? (In other words, do you obtain the same image regardless of the order in which you perform the transformations?) Justify your answer.

Question 39.
MATHEMATICAL CONNECTIONS
Point B (1, 4) is the image of B(3, 2) after a reflection in line c. Write an equation for line c.

Maintaining Mathematical Proficiency

Use the diagram to Íind the angle measure.

Question 40.
m∠AOC

Question 41.
m∠AOD

Question 42.
m∠BOE

Question 43.
m∠AOE

Question 44.
m∠COD

Question 45.
m∠EOD

Question 46.
m∠COE

Question 47.
m∠AOB

Question 48.
m∠COB

Question 49.
m∠BOD

### 4.3 Rotations

Exploration 1

Rotating a Triangle in a Coordinate Plane

Work with a partner:

a. Use dynamic geometry software to draw any triangle and label it ∆ABC.

b. Rotate the triangle 90° counterclockwise about the origin to from ∆A’B’C’.

c. What is the relationship between the coordinates of the vertices of ∆ABC and those of ∆A’B’C’?

d. What do you observe about the side lengths and angle measures of the two triangles?

Exploration 2

Rotating a Triangle in a Coordinate Plane

Work with a partner:
a. The point (x, y) is rotated 90° counterclockwise about the origin. Write a rule to determine the coordinates of the image of (x, y).
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to use previous established results in constructing arguments.

b. Use the rule you wrote in part (a) to rotate ∆ABC 90° counterclockwise about the origin. What are the coordinates of the vertices of the image. ∆A’B’C’?

c. Draw ∆A’B’C’. Are its side lengths the same as those of ∆ABC? Justify your answer.

Exploration 3

Rotating a Triangle in a Coordinate Plane

Work with a partner.

a. The point (x, y) is rotated 180° counterclockwise about the origin. Write a rule to
determine the coordinates of the image of (x, y). Explain how you found the rule.

b. Use the rule you wrote in part (a) to rotate ∆ABC (front Exploration 2) 180° counterclockwise about the origin. What are the coordinates of the vertices of the image, ∆A’B’C’?

Question 4.
How can you rotate a figure in a coordinate plane?

Question 5.
In Exploration 3. rotate A’B’C’ 180° counterclockwise about the origin. What are the coordinates of the vertices of the image. ∆A”B”C”? How are these coordinates related to the coordinates of the vertices of the original triangle, ∆ABC?

### Lesson 4.3 Rotations

Monitoring Progress

Question 1.
Trace ∆DEF and point P. Then draw a 50° rotation of ∆DEF about point P.

Question 2.
Graph ∆JKL with vertices J(3, 0), K(4, 3), and L(6, 0) and its image after a 90° rotation about the origin.

Question 3.
Graph $$\overline{R S}$$ from Example 3. Perform the rotation first, followed by the reflection. Does the order of the transformations matter? Explain.

Question 4.
WHAT IF?
In Example 3. $$\overline{R S}$$ is reflected in the x-axis and rotated 180° about the origin. Graph $$\overline{R S}$$ and its image after the composition.

Question 5.
Graph $$\overline{A B}$$ with endpoints A(- 4, 4) and B(- 1, 7) and its image after the composition.
Translation: (x, y) → (x – 2, y – 1)

Question 6.
Graph ∆TUV with vertices T(1, 2), U(3. 5), and V(6, 3) and its image after the composition.
Reflection: in the x-axis

Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself.

Question 7.
rhombus

Question 8.
octagon

Question 9.
right triangle

### Exercise 4.3 Rotations

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
When a point (a, b) is rotated counterclockwise about the origin. (a, b) → (b, – a) is the result of a rotation of _________ .

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.

What are the coordinates of the vertices of the image after a 90° counterclockwise rotation about the origin?

What are the coordinates of the vertices of the image after a 270° clockwise rotation about the origin?

What are the coordinates of the vertices of the image after turning the figure 90° to the left about the origin?

What are the coordinates of the vertices of the image after a 270° counterclockwise rotation about the origin?

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6. trace the polygon and point P. Then draw a rotation o the polygon about point P using the given number of degrees.

Question 3.
30°

Question 4.
80°

Question 5.
150°

Question 6.
130°

In Exercises 7-10. graph the polygon and its image after a rotation of the given number of degrees about the origin.

Question 7.
90°

Question 8.
180°

Question 9.
180°

Question 10.
270°

In Exercises 11-14, graph $$\overline{X Y}$$ with endpoints X(-3, 1) and Y(4, – 5) and its image after the composition.

Question 11.
Translation: (x, y) → (x, y + 2)

Question 12.
Translation: (x, y) → (x – 1, y + 1)

Question 13.
Reflection: in the y-axis

Question 14.
Reflection: in the line y = x

In Exercises 15 and 16, graph ∆LMN with vertices 2 L(1, 6), M(- 2, 4), and N(3, 2) and its image after the composition.

Question 15.
Translation: (x, y) → (x – 3, y + 2)

Question 16.
Reflection: in the x-axis

In Exercises 17-20, determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself.

Question 17.

Question 18.

Question 19.

Question 20.

REPEATED REASONING
In Exercises 21-24, select the angles of rotational symmetry for the regular polygon. Select all that apply.

(A) 30°        (B) 45°       (C) 60°         (D) 72°
(E) 90°         (F) 120°     (G) 144°      (H) 180°

Question 21.

Question 22.

Question 23.

Question 24.

ERROR ANALYSIS
In Exercises 25 and 26, the endpoints of $$\overline{C D}$$ are C(- 1, 1) and D(2, 3). Describe and correct the error in finding the coordinates of the vertices of the image after a rotation of 270° about the origin.

Question 25.

Question 26.

Question 27.
CONSTRUCTION
Follow these Steps to construct a rotation of ∆ABC by angle D around a point O. Use a compass and straightedge.

Step 1 Draw ∆ABC, ∠D, and O, the center of rotation.
Step 2 Draw $$\overline{O A}$$. Use the construction for copying an angle to copy ∠D at O. as shown. Then use distance OA and center O to find A’.
Step 3 Repeat Step 2 to find points B’ and C’. Draw ∆ A’B’C’.

Question 28.
REASONING
You enter the revolving door at a hotel.

a. You rotate the door 180°. what does this mean in the context of the situation? Explain.

b. You rotate the door 360°. What does this mean in the Context of the situation? Explain.

Question 29.
MATHEMATICAL CONNECTIONS
Use the graph of Y = 2X – 3.

a. Rotate the line 90°, 180° 270°, and 360° about the origin. Write the equation of the line for each image. Describe the relationship between the equation of the preimage and the equation of each image.
b. Do you think that the relationships you described in part (a) are true for any line? Explain your reasoning.

Question 30.
MAKING AN ARGUMENT
Your friend claims that rotating a figure by 180° is the same as reflecting a figure in the y-axis arid then reflecting it in the x-axis. Is your friend correct? Explain your reasoning.

Question 31.
DRAWING CONCLUSIONS
A figure only has point symmetry. How many times can you rotate the figure before it is back where it started?

Question 32.
ANALYZING RELATIONSHIPS
Is it possible for a figure to have 90° rotational symmetry but not 180° rotational symmetry? Explain your reasoning.

Question 33.
ANALYZING RELATIONSHIPS
Is it possible for a figure to have 180° rotational symmetry hut not 90° rotational symmetry? Explain your reasoning.

Question 34.
THOUGHT PROVOKING
Can rotations of 90°, 180°, 270°, and 360° be written as the composition of two reflections? Justify your answer.

Question 35.
USING AN EQUATION
Inside a kaleidoscope. two mirrors are placed next to each other to form a V. The angle between the mirrors determines the number of lines of symmetry in the image. Use the formula n(m∠1) = 180° to find the measure of ∠1, the angle between the mirrors, for the number n of lines of symmetry.

a.

b.

Question 36.
REASONING
Use the coordinate rules for counterclockwise rotations about the origin to write coordinate rules 11w clockwise rotations of 9o°. 180°, or 270° about the origin.

Question 37.
USING STRUCTURE
∆XYZ has vertices X(2, 5). Y(3, 1), and Z(0, 2). Rotate ∆XYZ 90° about the point P(- 2, – 1).

Question 38.
HOW DO YOU SEE IT?
You are finishing the puzzle. The remaining two pieces both have rotational symmetry.

a. Describe the rotational symmetry of Piece 1 and of Piece 2.

b. You pick up Piece 1. How many different ways can it fit in the puzzle?

c. Before putting Piece 1 into the puzzle, you connect it to Piece 2. Now how many ways can it fit in the puzzle? Explain.

Question 39.
USING STRUCTURE
A polar coordinate system locates a point in a plane by its distance from the origin O and by the measure of an angle with its vertex at the origin. For example, the point A(2, 30°) is 2 units from the origin and m∠XOA = 30°. What are the polar coordinates of the image of point A after a 90° rotation? a 180° rotation? a 270° rotation? Explain.

Maintaining Mathematical Proficiency

The figures are congruent. Name the corresponding angles and the corresponding sides.

Question 40.

Question 41.

### 4.1 – 4.3 Quiz

Graph quadrilateral ABCD with vertices A(- 4, 1), B(- 3, 3), C(0, 1), and D(- 2, 0) and its
image alter the translation.

Question 1.
(x, y) → (x + 4, y – 2)

Question 2.
(x, y) → (x – 1, y – 5)

Question 3.
(x, y) → (x + 3, y + 6)

Graph the polygon with the given vertices and its image after a reflection in the given line.

Question 4.
A(- 5, 6), B(- 7, 8), c(- 3, 11); x – axis

Question 5.
D(- 5, – 1), E(- 2, 1), F(- 1, – 3); y = x

Question 6.
J(- 1, 4), K(2, 5), L(5, 2), M(4, – 1); x = 3

Question 7.
P(2, – 4), Q(6, – 1), R(9, – 4), S(6, – 6); y = – 2

Graph ∆ABC with vertices A(2, – 1), B(5, 2), and C(8, – 2) and its image after the glide reflection.

Question 8.
Translation: (x, y) → (x, y + 6)
Reflection: in the y – axis

Question 9.
Translation: (x, y) → (x – 9, y)
Reflection: in the line y = 1

Determine the number of lines of symmetry for the figure.

Question 10.

Question 11.

Question 12.

Question 13.

Graph the polygon and its image after a rotation of the given number of degrees about the origin.

Question 14.
90°

Question 15.
270°

Question 16.
180°

Graph ∆LMN with vertices L(- 3, – 2), M (- 1, 1), and N(2, – 3) and its image after
the composition.

Question 17.
Translation: (x, y) → (x – 4, y + 3)

Question 18.
Reflection: in the y-axis

Question 19.
The figure shows a game in which the object is to create solid rows using the pieces given. Using only translations and rotations, describe the transformations for each piece at the top that will form two solid rows at the bottom.

### 4.4 Congruence and Transformations

Exploration 1

Reflections in Parallel Lines

Work with a partner. Use dynamic geometry software to draw any scalene triangle and label it ∆ABC.

a. Draw an line . Reflect ∆ ABC in to form ∆A’B’C’.

b. Draw a line parallel to . Reflect ∆A’B’C’ in the new line to form ∆A”B”C”.

c. Draw the line through point A that is perpendicular to . What do you notice?

d. Find the distance between points A and A”. Find the distance between the two parallel lines. What do You notice?

e. Hide ∆A’B’C’. Is there a single transformation that maps ∆ABC to ∆A”B”C”? Explain.

f. Make conjectures based on your answers in parts (c)-(e). Test our conjectures by changing ∆ABC and the parallel lines.
CONSTRUCTING VIABLE ARGUMENTS
To be proficient in math, you need to make conjectures and justify your conclusions.

Exploration 2

Reflections in Intersecting Lines

Work with a partner: Use dynamic geometry software to draw any scalene triangle and label it ∆ABC.

a. Draw an line . Reflect ∆ABC in to form ∆A’B’C’.

b. Draw any line so that angle EDF is less than or equal to 90°. Reflect ∆A’B’C’ in to form ∆A”B”C”.

c. Find the measure of ∠EDF. Rotate ∆ABC counterclockwise about point D using an angle twice the measure of ∠EDF.

d. Make a conjecture about a figure reflected in two intersecting lines. Test your conjecture by changing ∆ABC and the lines.

Question 3.
What conjectures can you make about a figure reflected in two lines?

Question 4.
Point Q is reflected in two parallel lines, and . to form Q’ and The distance from to is 3.2 inches. What is the distance QQ”?

### Lesson 4.4 Congruence and Transformations

Monitoring Progress

Question 1.
Identify any congruent figures in the coordinate plane. Explain.

Question 2.
In Example 2. describe another congruence transformation that maps ▱ABCD to ▱EFGH.

Question 3.
Describe a congruence transformation that maps △JKL to △MNP.

Use the figure. The distance between line k and line m is 1.6 centimeters.

Question 4.
The preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure to the green figure.

Question 5.
What is the relationship between $$\overline{P P’}$$ and line k? Explain.

Question 6.
What is the distance between P and P”?

Question 7.
In the diagram. the preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure onto the green figure.

Question 8.
A rotation of 76° maps C to C’. To map C to C’ Using two reflections, what is the measure of the angle formed by the intersecting lines of reflection?

### Exercise 4.4 Congruence and Transformations

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
Two geometric figures are __________ if and only if there is a rigid motion or a composition of rigid motions that moves one of the figures onto the other.

Question 2.
VOCABULARY
Why is the term congruence transformation used to refer to a rigid motion?

Monitoring Progress and Modeling with Mathematics

In Exercises 3 and 4, identify an congruent figures in the coordinate plane. Explain.

Question 3.

Question 4.

In Exercises 5 and 6, describe a congruence transformation that maps the blue preimage to the green image.

Question 5.

Question 6.

In Exercises 7-10. determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning.

Question 7.
Q(2, 4), R(5, 4), S(4, 1) and T(6, 4), U(9, 4), V(8, 1)

Question 8.
W(- 3, 1), X(2, 1), Y(4, -,4),,Z(- 5, – 4) and C(- 1, – 3) D(- 1, 2), E(4, 4), F(4, – 5)

Question 9.
J(1, 1), K(3, 2), L(4, 1) and M(6, 1), N(5, 2), P(2, 1)

Question 10.
A(0, 0), B(1, 2), C(4, 2), D(3, 0) and E(0, – 5), F( – 1, – 3), G(- 4, – 3), H(- 3, – 5)

In Exercises 11-14, k || m, ∆ABC is reflected in line k, and ∆A’B’C” is reflected in line in.

Question 11.
A translation maps ∆ABC onto which triangle?

Question 12.
Which lines are perpendicular to $$\overline{A A”}$$?

Question 13.
If the distance between k and m is 2.6 inches. what is the length of $$\overline{C C”}$$?

Question 14.
Is the distance from B’ to in the same as the distance from B” to m? Explain.

In Exercises 15 and 16, find the angle of rotation that maps A onto A”.

Question 15.

Question 16.

Question 17.
ERROR ANALYSIS
Describe and correct the error in describing the congruence transformation.

Question 18.
ERROR ANALYSIS
Describe and correct the error in using the Reflections in Intersecting Lines Theorem

In Exercises 19 – 22, find the measure of the acute or right angle formed by intersecting lines so that C can be mapped to C’ using two reflections.

Question 19.
A rotation of 84° maps C to C’.

Question 20.
A rotation of 24° maps C to C’.

Question 21.
The rotation (x, y) → (- x, – y) maps C to C’.

Question 22.
The rotation (x, y) → (y, – x) maps C to C’.

Question 23.
REASONING
Use the Reflection in Parallel Lines Theorem (Theorem 4.2) to explain how you can make a glide reflection using three reflections. How are the lines of reflection related?

Question 24.
DRAWING CONCLUSIONS
The pattern shown is called a tessellation.

a. What transformations did the artist use when creating this tessellation?

b. Are the individual figures in the tessellation congruent? Explain your reasoning.

CRITICAL THINKING
In Exercises 25-28, tell whether the
statement is away, sometime or never true. Explain your reasoning.

Question 25.
A Congruence transformation changes the size of a figure.

Question 26.
If two figures are Congruent, then there is a rigid motion or a composition of rigid motions that maps one figure onto the other.

Question 27.
The composition of two reflections results in the same image as a rotation.

Question 28.
A translation results in the same image as the composition of two reflections.

Question 29.
REASONING
During a presentation, a marketing representative uses a projector so everyone in the auditorium can view the advertisement. Is this projection a congruence transformation? Explain your reasoning.

Question 30.
HOW DO YOU SEE IT?
What type of congruence transformation can be used to verify each statement about the stained glass window?

a. Triangle 5 is congruent to Triangle 8.

b. Triangle 1 is congruent to Triangle 4.

c. Triangle 2 is congruent to Triangle 7.

d. Pen1aon 3 is congruent to Pentagon 6.

Question 31.
PROVING A THEOREM
Prove the Reflections in Parallel Lines Theorem (Theorem 4.2).

Given A reflection in line l maps $$\overline{J K}$$ to $$\overline{J’ K’}$$.
a reflection in line in maps $$\overline{J’ K’}$$ to $$\overline{J” K”}$$.
and l || m.
Prove a. $$\overline{K K”}$$ is perpendicular to l and m. b. KK” = 2d, where d is the distance between l and m.

Question 32.
THOUGHT PROVOKING
A tessellation is the covering of a plane with congruent figures so that there are no gaps or overlaps (see Exercise 24). Draw a tessellation that involves two or more types of transformations. Describe the transformations that are used to create the tessellation.

Question 33.
MAKING AN ARGUMENT
$$\overline{P Q}$$, with endpoints P(1, 3) and Q(3, 2). is reflected in the y-axis. The image $$\overline{P’ Q’}$$ is then reflected in the x-axis to produce the image $$\overline{P” Q”}$$. One classmate says that $$\overline{P Q}$$ is mapped to $$\overline{P” Q”}$$ by the translation (x, y) → (x – 4, y – 5). Another classmate says that $$\overline{P Q}$$ is mapped to $$\overline{P” Q”}$$ by a (2 • 90)°, or 180°, rotation about the origin. Which classmate is correct? Explain your reasoning.

Question 34.
CRITICAL THINKING
Does the order of reflections for a composition of two reflections in parallel lines matter? For example, is reflecting ∆XYZ in line l and then its image in line in the same as reflecting ∆XYZ in line in and then its image in line l ?

CONSTRUCTION
In Exercises 35 and 36. copy the figure. Then use a compass and straightedge to construct two lines of reflection that produce a composition of reflections resulting in the same image as the given transformation.

Question 35.
Translation: ∆ABC → ∆A”B”C”

Question 36.
Rotation about P: ∆XYZ → ∆X”Y”Z”

Maintaining Mathematical Proficiency

Solve the equation. Check your solution.

Question 37.
5x + 16 = – 3x

Question 38.
12 + 6m = 2m

Question 39.
4b + 8 = 6b – 4

Question 40.
7w – 9 = 13 – 4w

Question 41.
7(2n + 11) = 4n

Question 42.
-2(8 – y) = – 6y

Question 43.
Last year. the track team’s yard sale earned $500. This year. the yard sale earned$625. What is the percent of increase?

### 4.5 Dilations

Exploration 1

Dilating a Triangle in a Coordinate Plane

Work with a partner: Use dynamic geometry software to draw any triangle and label
it ∆ABC.

a. Dilate ∆ABC using a scale factor of 2 and a center of dilation at the origin to form ∆A’B’C’. Compare the coordinates, side lengths. and angle measures of ∆ABC and ∆A’B’C’.

b. Repeat part (a) using a scale factor of $$\frac{1}{2}$$
LOOKING FOR STRUCTURE
To be proficient in math, you need to look closely to discern a pattern or structure.

c. What do the results of parts (a) and (b) suggest about the coordinates, side lengths, and angle measures of the image of ∆ABC after a dilation with a scale factor of k?

Exploration 2

Dilating Lines in a Coordinate Plane

Work with a partner. Use dynamic geometry software to draw that passes through the origin and that does not pass through the origin.

a. Dilate using a scale factor of 3 and a center of dilation at the origin. Describe the image.

b. Dilate using a scale factor of 3 and a center of dilation at the origin. Describe the image.

c. Repeat parts (a) and (b) using a scale factor of $$\frac{1}{4}$$

d. What do you notice about dilations of lines passing through the center of dilation and dilations of lines not passing through the center of dilation?

Question 3.
What does it mean to dilate a figure?

Question 4.
Repeat Exploration 1 using a center of dilation at a point other than the origin.

### Lesson 4.5 Dilations

Monitoring Progress

Question 1.
In a dilation. CP’ = 3 and CP = 12. Find the scale factor. Then tell whether the dilation is a reduction or an enlargement.

Graph ∆PQR and its image alter a dilation with scale factor k.

Question 2.
P(- 2, – 1), Q(- 1, 0), R(0, – L); k = 4

Question 3.
P(5, – 5), Q( 10, – 5), R( 10, 5); k = 0.4

Question 4.
Graph ∆PQR with vertices P(1, 2), Q(3, 1). and R( 1, – 3) and its image after a dilation with a scale factor of – 2.

Question 5.
Suppose a figure containing the origin is dilated. Explain why the corresponding point in the image of the figure is also the origin

Question 6.
An optometrist dilates the pupils of a patient’s eyes to get a better look at the back of the eyes. A pupil dilates from 4.5 millimeters to 8 millimeters. What is the scale factor of this dilation?

Question 7.
The image of a spider seen through the magnifying glass in Example 6 is shown at the left. Find the actual length of the spider.

### Exercise 4.5 Dilations

Vocabulary and Core Concept Check

Question 1.
COMPLETE THE SENTENCE
If P(x. y) is the preimage of a point, then its image after a dilation centered at the origin (0, 0) with scale factor k is the point _________.

Question 2.
WHICH ONE DOESNT BELONG?
Which scale factor does not belong with the other three? Explain your reasoning.
$$\frac{5}{4}$$ 60% 115% 2

Monitoring Progress Modeling with Mathematics

In Exercises 3-6. find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement.

Question 3.

Question 4.

Question 5.

Question 6.

CONSTRUCTION
In Exercises 7-10. copy the diagram. Then use a compass and straightedge to construct a dilation of ∆LMN with the given center and scale factor k.

Question 7.
Center C, k = 2

Question 8.
Center P, k = 3

Question 9.
Center M, k = $$\frac{1}{2}$$

Question 10.
Center C. k = 25%

CONSTRUCTION
In Exercises 11-14, copy the diagram. Then use a coin pass and straightedge to construct a dilation of quadrilateral RSTU with the given center and scale factor k.

Question 11.
Center C, k = 3

Question 12.
Center P, k = $$\frac{1}{3}$$

Question 13.
Center P, k = 0.25

Question 14.
Center C, k = 75%

In Exercises 15-18, graph the polygon and its image after a dilation with scale factor k.

Question 15.
X(6, – 1), Y(- 2, – 4), Z(1, 2); k = 3

Question 16.
A(0, 5), B(- 10, – 5), C(5, – 5); k = 12o%

Question 17.
T(9, – 3), U(6, 0), V(3, 9), W(0. 0); k = $$\frac{2}{3}$$

Question 18.
J(4, 0), K(- 8, 4), L(0, – 4), M(12, – 8);k = 0. 25

In Exercises 19-22, graph the polygon and its image after a dilation with scale factor k.

Question 19.
B(- 5, – 10), C(- 10, 15), D(0, 5); k = 3

Question 20.
L(0, 0), M(- 4, 1), N(- 3, – 6); k = – 3

Question 21.
R(- 7, – 1), S(2, 5), T(- 2, – 3), U(- 3,- 3); k = – 4

Question 22.
W(8, – 2), X(6, 0), Y(- 6, 4), Z(- 2, 2); k = – 0.5

ERROR ANALYSIS
In Exercises 23 and 24, describe and correct the error in finding the scale factor of the dilation.

Question 23.

Question 24.

In Exercises 25-28, the red figure is the image of the blue figure after a dilation with center C. Find the scale factor of the dilation. Then find the value of the variable.

Question 25.

Question 26.

Question 27.

Question 28.

Question 29.
FINDING A SCALE FACTOR
You receive wallet-sized photos of your school picture. The photo is 2.5 inches by 3.5 inches. You decide to dilate the photo to 5 inches by 7 inches at the store. What is the scale factor of this dilation?

Question 30.
FINDING A SCALE FACTOR
Your visually impaired friend asked you to enlarge your notes from class so he can study. You took notes on 8.5-inch by 11-inch paper. The enlarged copy has a smaller side with a length of 10 inches. What is the scale factor of this dilation?

In Exercises 31-34, you are using a magnifying glass. Use the length of the insect and the magnification level to determine the length of the image seen through the magnifying glass.

Question 31.
emperor moth
Magnification: 5×

Question 32.
Magnification: 10×

Question 33.
dragonfly
Magnification: 20×

Question 34.
carpenter ant
Magnification: 15×

Question 35.
ANALYZING RELATIONSHIPS
Use the given actual and magnified lengths to determine which of the following insects were looked at using the same magnifying in glass. Explain your reasoning.

Question 36.
THOUGHT PROVOKING
Draw ∆ABC and ∆A’B’C’ so that ∆A’B’C’ is a dilation of ∆ABC. Find the center of dilation and explain how you found it.

Question 37.
REASONING
Your friend prints a 4-inch by 6-inch photo for you from the school dance. All you have is an 8-inch by 10-inch frame. Can you dilate the photo to fit the frame? Explain your reasoning.

Question 38.
HOW DO YOU SEE IT?
Point C is the center of dilation of the images. The scale factor is $$\frac{1}{3}$$. Which figure is the original figure? Which figure is the dilated figure? Explain your reasoning.

Question 39.
MATHEMATICAL CONNECTIONS
The larger triangle is a dilation of the smaller triangle. Find the values of x and y.

Question 40.
WRITING
Explain why a scale factor of 2 is the same as 200%.

In Exercises 41-44, determine whether the dilated figure or the original figure is closer to the center of dilation. Use the given location of the center of dilation and scale factor k.

Question 41.
Center of dilation: inside the figure; k = 3

Question 42.
Center of dilation: inside the figure; k = $$\frac{1}{2}$$

Question 43.
Center of dilation: outside the figure; k = 120%

Question 44.
Center of dilation: outside the figure; k = 0. 1

Question 45.
ANALYZING RELATIONSHIPS
Dilate the line through 0(0, 0) and A(1, 2) using a scale factor of 2.
a. What do you notice about the lengths of $$\overline{O’ A’}$$ and $$\overline{O A}$$?
b. What do you notice about

Question 46.
ANALYZING RELATIONSHIPS
Dilate the line through A(0, 1) and B( 1, 2) using a scale factor of $$\frac{1}{2}$$.

a. What do you notice about the lengths of $$\overline{A’ B’}$$ and $$\overline{A B}$$?

b. What do you notice about ?

Question 47.
ATTENDING TO PRECISION
You are making a blueprint of your house. You measure the lengths of the walls of your room to be 11 feet by 12 feet. When you draw your room on the blueprint, the lengths of the walls are 8.25 inches by 9 inches. What scale factor dilates your room to the blueprint?

Question 48.
MAKING AN ARGUMENT
Your friend claims that dilating a figure by 1 is the same as dilating a figure by – 1 because the original figure will not be enlarged or reduced. Is your friend correct? Explain your reasoning.

Question 49.
USING STRUCTURE
Rectangle WXYZ has vertices W(- 3, – 1), X(- 3, 3), Y(5, 3), and Z(5, – 1).
a. Find the perimeter and area of the rectangle.
b. Dilate the rectangle using a scale factor of 3. Find the perimeter and area of the dilated rectangle. Compare with the original rectangle. What do you notice?
c. Repeat part (b) using a scale factor of $$\frac{1}{4}$$.
d. Make a conjecture for how the perimeter and area change when a figure is dilated.

Question 50.
REASONING
You put a reduction of a page on the original page. Explain why there is a point that is in the same place on both pages.

Question 51.
REASONING
∆ABC has vertices A(4, 2), B(4, 6), and C(7, 2). Find the coordinates of the vertices of the image alter a dilation with center (4, 0) and a scale factor of 2.

Maintaining Mathematical Proficiency

The vertices of ∆ABC are A(2,- 1), B(0, 4), and C(- 3, 5). Find the coordinates of the vertices of the image after the translation.

Question 52.
(x, y) → (x, y – 4)

Question 53.
(x, y) → (x – 1, y + 3)

Question 54.
(x, y) → (x + 3, y – 1)

Question 55.
(x, y) → (x – 2, y)

Question 56.
(x, y) → (x + 1, y – 2)

Question 57.
(x, y) → (x – 3y + 1)

### 4.6 Similarity and Transformations

Exploration 1

Dilations and similarity

Work with a partner.

a. Use dynamic geometry software to draw any triangle and label it ∆ABC.

b. Dilate the triangle using a scale factor of 3. Is the image similar to the original triangle? Justify your answer.
ATTENDING TO PRECISION
To be proficient in math, you need to use clear definitions in discussions with others and in your own reasoning.

Exploration 2

Rigid Motions and Similarity

Work with a partner.
a. Use dynamic geometry software to draw any triangle.

b. Copy the triangle and translate it 3 units left and 4 units up. Is the image similar to the original triangle? Justify your answer.

c. Reflect the triangle in the y-axis. Is the image similar to the original triangle? Justify your answer.

d. Rotate the original triangle 90° counterclockwise about the origin. Is the image similar to the original triangle? Justify your answer.

Question 3.
When a figure is translated, reflected, rotated, or dilated in the plane, is the image
always similar to the original figure? Explain your reasoning.

Question 4.
A figure undergoes a composition of transformations. which includes translations.
reflections, rotations, and dilations. Is the image similar to the original figure?

### Lesson 4.6 Similarity and Transformations

Monitoring Progress

Question 1.
Graph $$\overline{C D}$$ with endpoints C(- 2, 2) and D(2, 2) and its image after the similarity transformation.
Dilation: (x, y) → $$\left(\frac{1}{2} x, \frac{1}{2} y\right)$$

Question 2.
Graph ∆FGH with vertices F(1, 2), G(4, 4), and H(2, 0) and its image after the similarity transformation.
Reflection: in the x-axis
Dilation: (x, y) → (1.5x, 1.5y)

Question 3.
In Example 2, describe another similarity transformation that maps trapezoid PQRS to trapezoid WXYZ.

Question 4.

Question 5.
Prove that ∆JKL is similar to ∆MNP.
Given Right isosceles ∆JKL with leg length t, right isosceles ∆MNP with leg length ν,
$$\overline{L J}$$ || $$\overline{P M}$$
Prove ∆JKL is similar to ∆MNP.

### Exercise 4.6 Similarity and Transformations

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
What is the difference between similar figures and congruent figures?

Question 2.
COMPLETE THE SENTENCE
A transformation that produces a similar figure. such as a dilation.
is called a _________ .

Monitoring Progress and Modeling with Mathematics

In Exercises 3-6, graph ∆FGH with vertices F(- 2, 2), G(- 2, – 4), and H(- 4, – 4) and its image after the similarity transformation.

Question 3.
Translation: (x, y) → (x + 3, y + 1)
Dilation: (x, y) → (2x, 2y)

Question 4.
Dilation: (x, y) → $$\left(\frac{1}{2} x, \frac{1}{2} y\right)$$
Reflection: in the y-axis

Question 5.
Dilation: (x, y) → (3x, 3y)

Question 6.
Dilation: (x, y) → $$\left(\frac{3}{4} x, \frac{3}{4} y\right)$$
Reflection: in the x-axis

In Exercises 7 and 8. describe a similarity transformation that maps the blue preimage to the green image.

Question 7.

Question 8.

In Exercises 9-12, determine whether the polygons with the given vertices are similar. Use transformations to explain your reasoning.
Question 9.
A6, 0), B(9, 6), C(12, 6) and D(0, 3), E( 1, 5), F(2. 5)

Question 10.
Q(- 1, 0), R(- 2, 2), S(1, 3), T(2, 1) and W(0, 2), X(4, 4), Y(6, – 2), Z(2, – 4)

Question 11.
G(- 2, 3), H(4, 3), I(4, 0) and J(1, 0), K(6, – 2), L(1, – 2)

Question 12.
D(- 4, 3), E(- 2, 3), F(- 1, 1), G(- 4, 1) and L(1, – 1), M(3, – 1), N(6, – 3), P(1, – 3)

In Exercises 13 and 14, prove that the figures are similar.

Question 13.
Given Right isosceles ∆ABC with leg length j.
right isosceles ∆RST with leg length k.
$$\overline{C A}$$ || $$\overline{R T}$$
Prove ∆ABC is similar to ∆RST.

Question 14.
Given Rectangle JKLM with side lengths x and y, rectangle QRST with side lengths 2x and 2y
Prove Rectangle JKLM is similar to rectangle QRST.

Question 15.
MODELING WITH MATHEMATICS
Determine whether the regular-sized stop sign and the stop sign sticker are similar. Use transformations to explain your reasoning.

Question 16.
ERROR ANALYSIS
Describe and correct the error in comparing the figures.

Question 17.
MAKING AN ARGUMENT
A member of the homecoming decorating committee gives a printing company a banner that is 3 inches by 14 inches to enlarge. The committee member claims the banner she receives is distorted. Do you think the printing company distorted the image she gave it? Explain.

Question 18.
HOW DO YOU SEE IT?
Determine whether each pair of figures is similar. Explain your reasoning.
a.

b.

Question 19.
ANALYZING RELATIONSHIPS
Graph a polygon in a coordinate plane. Use a similarity transformation involving a dilation (where k is a whole number) and a translation to graph a second polygon. Then describe a similarity transformation that maps the second polygon onto the first.

Question 20.
THOUGHT PROVOKING
Is the composition of a rotation and a dilation commutative? (In other words. do you obtain the same image regardless of the order in which you perform the transformations?) Justify your answer.

Question 21.
MATHEMATICAL CONNECTIONS
Quadrilateral JKLM is mapped to quadrilateral J’K’L’M’ using the dilation (x, y) → $$\left(\frac{3}{2} x, \frac{3}{2} y\right)$$. Then quadrilateral J’K’L’M is mapped to quadrilateral J”K”L”M” using the translation (x, y) → (x + 3, y – 4). The vertices of quadrilateral J’K’L’M’ are J(- 12, 0), K(- 12, 18), L(- 6, 18), and M(- 6, 0), Find the coordinates of the vertices of quadrilateral JKLM and quadrilateral J”K”L”M”. Are quadrilateral JKLM and quadrilateral J”K”L”M” similar? Explain.

Question 22.
REPEATED REASONING
Use the diagram.

a. Connect the midpoints of the sides of ∆QRS to make another triangle. Is this triangle similar to ∆QRS? Use transformations to support your answer.

b. Repeat part (a) for two other triangles. What conjecture can you make?

Maintaining Mathematical Proficiency

Classify the angle as acute, obtuse, right, or straight.

Question 23.

Question 24.

Question 25.

Question 26.

### Transformations Chapter Review

#### 4.1 Translations

Graph ∆XYZ with vertices X(2, 3), Y(- 3, 2), and Z(- 4, – 3) and its image after the translation.
Question 1.
(x, y) → (x, y + 2)

Question 2.
(x, y) → (x – 3, y)

Question 3.
(x, y) → (x + 3y – 1)

Question 4.
(x, y) → (x + 4, y + 1)

Graph ∆PQR with vertices P(0, – 4), Q(1, 3), and R(2, – 5) and its image after the composition.

Question 5.
Translation: (x, y) → (x + 1, y + 2)
Translation: (x, y) → (x – 4, y + 1)

Question 6.
Translation: (x, y) → (x, y + 3)
Translation: (x, y) → (x – 1, y + 1)

#### 4.2 Reflections

Graph the polygon and its image after a reflection in the given line.

Question 7.
x = 4

Question 8.
y = 3

Question 9.
How many lines of symmetry does the figure have?

#### 4.3 Rotations

Question 10.
A(- 3, – 1), B(2, 2), C(3, – 3); 90°

Question 11.
W(- 2, – 1), X(- 1, 3), Y(3, 3), Z(3, – 3); 180°

Question 12.
Graph $$\overline{X Y}$$ with endpoints X(5, – 2) and Y(3, – 3) and its image after a reflection in the x-axis and then a rotation of 270° about the origin.

Determine whether the figure has rotational symmetry. If so, describe any rotations that map the figure onto itself.

Question 13.

Question 14.

#### 4.4 Congruence and Transformations

Describe a congruence transformation that maps ∆DEF to ∆JKL.

Question 15.
D(2, – 1), E(4, 1), F(1, 2) and J(- 2, – 4), K(- 4, – 2), L(- 1, – 1)

Question 16.
D(- 3, – 4), E(- 5, – 1), F(- 1, 1) and J(1, 4), K(- 1, 1), L(3, – 1)

Question 17.
Which transformation is the same as reflecting an object in two Parallel lines? in two intersecting lines?

#### 4.5 Dilations

Graph the triangle and its image after a dilation with scale factor k.

Question 18.
P(2, 2), Q(4, 4), R(8, 2); k = $$\frac{1}{2}$$

Question 19.
X(- 3, 2), Y(2, 3), Z(1, – 1); k = – 3

Question 20.
You are using a magnifying glass that shows the image of an object that is eight times the object’s actual size. The image length is 15.2 centimeters. Find the actual length of the object.

#### 4.6 Similarity and Transformations

Describe a similarity transformation that maps ∆ABC to ∆RST.

Question 21.
A(1, 0), B(- 2, – 1), C(- 1, – 2) and R(- 3, 0), S(6, – 3), T(3, – 6)

Question 22.
A(6, 4), B(- 2, 0), C(- 4, 2) and R(2, 3), S(0, – 1), T(1, – 2)

Question 23.
A(3, – 2), B(0, 4), C(- 1, – 3) and R(- 4, – 6), S(8, 0), T(- 6, 2)

### Transformations Test

Graph ∆RST with vertices R(- 4, 1), S(- 2, 2), and T(3, – 2) and its image after the translation.

Question 1.
(x, y) → (x – 4, y + 1)

Question 2.
(x, y) → (x + 2, y – 2)

Graph the polygon with the given vertices and its image after a rotation of the given number of degrees about the origin.

Question 3.
D(- 1, – 1), E(- 3, 2), F(1, 4); 270°

Question 4.
J(- 1, 1), K(3, 3), L(4, – 3), M(0, – 2); 90°

Determine whether the polygons with the given vertices are congruent or similar. Use transformations to explain your reasoning.

Question 5.
Q(2, 4), R(5, 4), S(6, 2), T(1, 2) and W(6, – 12), X(15, – 12), Y(18, – 6), Z(3, -,6)

Question 6.
A(- 6, 6), B(- 6, 2), C(- 2, – 4) and D(9, 7), E(5, 7), F(- 1, 3)

Determine whether the object has line symmetry and whether it has rotational symmetry.
Identify all lines of symmetry and angles of rotation that map the figure onto itself.

Question 7.

Question 8.

Question 9.

Question 10.
Draw a diagram using a coordinate plane. two parallel lines, and a parallelogram that demonstrates the Reflections in Parallel Lines Theorem (Theorem 4.2).

Question 11.
A rectangle with vertices W(- 2, 4), X(2, 4), Y(2, 2), and Z(- 2, 2) is reflected in the y-axis. Your friend says that the image. rectangle W’X’ Y’Z. is exactly the same as the preimage. Is your friend correct? Explain your reasoning.

Question 12.
Write a composition of transformations that maps ∆ABC Onto ∆CDB in the tesselation shown. Is the composition a congruence transformation? Explain your reasoning.

Question 13.
There is one slice of a large pizza and one slice of a small pizza in the box.

a. Describe a similarity transformation that maps pizza slice ABC to pizza slice DEF.

b. What is one possible scale factor for a medium slice of pizza? Explain your reasoning.

Question 14.
The original photograph shown is 4 inches by 6 inches.

a. What transformations can you use to produce the new photograph?

b. You dilate the original photograph b a scale factor of $$\frac{1}{2}$$. What are the dimensions of the new photograph?

c. YOU have a frame that holds photos that are 8.5 inches by 11 inches. Can you dilate the original photograph to fit the frame? Explain your reasoning.

### Transformations Cumulative Assessment

Question 1.
Which composition 0f transformations maps ∆ABC to ∆DEF?

(A) Rotation: 90° counterclockwise about the origin
Translation: (x, y) → (x + 4, y – 3)

(B) Translation: (x, y) → (x – 4, y – 3)
Rotation: 90° counterclockwise about the origin

(C) Translation: (x, y) → (x + 4, y – 3)
Rotation: 90° counterclockwise about the origin

(b) Rotation: 90° counterclockwise about the origin
Translation: (x, y) → (x – 4, y – 3)

Question 2.
Use the diagrams to describe the steps you would take to construct a line perpendicular to line m through point P. which is not on line m.

Question 3.
Your friend claims that she can find the perimeter of the school crossing sign without using the Distance Formula. Do you support your friend’s claim? Explain your reasoning.

Question 4.
Graph the directed line segment ST with endpoints S(- 3, – 2) and T(4, 5). Then find the coordinates of point P along the directed line segment ST so that the ratio of SP to PT is 3 to 4.

Question 5.

a. Write a composition of transformations that maps quadrilateral WXYZ to

Question 6.
Which equation represents the line passing through the point (- 6, 3) that is parallel to
the line y = – $$\frac{1}{3}$$x – 5?
(A) y = 3x + 21
(B) y = –$$\frac{1}{3}$$x – 5
(C) y = 3x – 15
() y = –$$\frac{1}{3}$$x + 1

Question 7.
Which scale factor(s) would create a dilation of $$\overline{A B}$$ that is shorter than $$\overline{A B}$$? Select all that apply.

Question 8.
List one possible set of coordinates of the vertices of quadrilateral ABCD for each description.
a. A reflection in the y-axis maps quadrilateral ABCD onto itself.