Big Ideas Math Geometry Answers Chapter 4

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 1
Answer:

Question 2.
Big Ideas Math Geometry Answers Chapter 4 Transformations 2
Answer:

Question 3.
Big Ideas Math Geometry Answers Chapter 4 Transformations 3
Answer:

Question 4.
Big Ideas Math Geometry Answers Chapter 4 Transformations 4
Answer:

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

Question 5.
Big Ideas Math Geometry Answers Chapter 4 Transformations 5
Answer:

Question 6.
Big Ideas Math Geometry Answers Chapter 4 Transformations 6
Answer:

Question 7.
Big Ideas Math Geometry Answers Chapter 4 Transformations 7
Answer:

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)
Answer:

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

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

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

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

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

4.1 Translations

Exploration 1

Translating a Triangle in a Coordinate Plane

Big Ideas Math Geometry Answers Chapter 4 Transformations 8
Work with a partner.
a. Use dynamic geometry software to draw any triangle and label it ∆ABC.
Answer:

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.
Answer:

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

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

Exploration 2

Translating a Triangle in a Coordinate Plane

Big Ideas Math Geometry Answers Chapter 4 Transformations 9
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).
Big Ideas Math Geometry Answers Chapter 4 Transformations 10
Answer:

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’?
Answer:

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

Exploration 3

Comparing Angles of Translations

Work with a partner.

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

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

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

Communicate Your Answer

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

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?
Answer:

Lesson 4.1 Translations

Monitoring Progress

Question 1.
Name the vector and write its component form.
Big Ideas Math Geometry Answers Chapter 4 Transformations 11
Answer:

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

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

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).
Answer:

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)
Answer:

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)
Answer:

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.
Answer:

Exercise 4.1 Translations

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
Name the preimage and image of the transformation ∆ABC – ∆A’B’C’.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 1

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

Monitoring Progress and Modeling with Mathematics

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

Question 3.
Big Ideas Math Geometry Answers Chapter 4 Transformations 12
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 3

Question 4.
Big Ideas Math Geometry Answers Chapter 4 Transformations 13
Answer:

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)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 5

Question 6.
(5, – 1)
Answer:

Question 7.
(- 3, – 7)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 7

Question 8.
(- 2, – 4)
Answer:

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

Question 9.
P'(0, 1)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 9

Question 10.
P'(- 4, 8)
Answer:

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

Question 11.
Big Ideas Math Geometry Answers Chapter 4 Transformations 14
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 11

Question 12.
Big Ideas Math Geometry Answers Chapter 4 Transformations 15
Answer:

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

Question 13.
What is the image of A(2, 6)?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 13

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

Question 15.
What is the preimage of C'(- 3, – 10)?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 15

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

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)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 17

Question 18.
(x, y) → (x + 9, y – 2)
Answer:

Question 19.
(x, y) → (x – 2, y – 5)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 19

Question 20.
(x, y) → (x – 1, y + 3)
Answer:

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)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 21

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

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

Question 23.
Big Ideas Math Geometry Answers Chapter 4 Transformations 16
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 23

Question 24.
Big Ideas Math Geometry Answers Chapter 4 Transformations 17
Answer:

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).
Big Ideas Math Geometry Answers Chapter 4 Transformations 18
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 25

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 19
Answer:

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 20
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?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 27

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.
Answer:

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.
Answer:

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

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 21
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 29

Question 30.
Big Ideas Math Geometry Answers Chapter 4 Transformations 22
Answer:

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’?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 31

Question 32.
HOW DO YOU SEE IT?
Which two figures represent a translation? Describe the translation.
Big Ideas Math Geometry Answers Chapter 4 Transformations 23
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 33

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’.
Answer:

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

Question 35.
PROVING A THEOREM
Prove the Composition Theorem (Theorem 4.1).
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 35

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

b. Corresponding Angles Converse (Theorem 3.5)
Answer:

Question 37.
WRITING
Explain how to use translations to draw a rectangular prism.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 37

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.
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 39

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.
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 41

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.
Answer:

Maintaining Mathematical Proficiency

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

Question 43.
Big Ideas Math Geometry Answers Chapter 4 Transformations 24
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 43

Question 44.
Big Ideas Math Geometry Answers Chapter 4 Transformations 25
Answer:

Question 45.
Big Ideas Math Geometry Answers Chapter 4 Transformations 26
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 45

Question 46.
Big Ideas Math Geometry Answers Chapter 4 Transformations 27
Answer:

Simplify the expression.

Question 47.
– (- x)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 47

Question 48.
– (x + 3)
Answer:

Question 49.
x – (12 – 5x)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.1 Question 49

Question 50.
x – (- 2x + 4)
Answer:

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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 28

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

b. Place a reflective device on line in.
Answer:

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.
Answer:

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

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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 29
a. Reflect ∆ABC in the y-axis to form ∆A’B’C’.
Answer:

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.
Answer:

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

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

Communicate Your Answer

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

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
Answer:

Question 2.
x = – 3
Answer:

Question 3.
y = 2
Answer:

Question 4.
y = – 1
Answer:

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.
Answer:

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

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

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

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

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.
Answer:

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

Determine the number of lines of symmetry for the figure.

Question 12.
Big Ideas Math Answers Geometry Chapter 4 Transformations 30
Answer:

Question 13.
Big Ideas Math Answers Geometry Chapter 4 Transformations 31
Answer:

Question 14.
Big Ideas Math Answers Geometry Chapter 4 Transformations 32
Answer:

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

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

Exercise 4.2 Reflections

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
A glide reflection is a combination of which two transformations?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 1

Question 2.
WHICH ONE DOESN’T BELONG?
Which transformation does not belong with the other three? Explain your reasoning.
Answer:
Big Ideas Math Answers Geometry Chapter 4 Transformations 33

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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 34
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 3

Question 4.
Big Ideas Math Answers Geometry Chapter 4 Transformations 35
Answer:

Question 5.
Big Ideas Math Answers Geometry Chapter 4 Transformations 36
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 5

Question 6.
Big Ideas Math Answers Geometry Chapter 4 Transformations 37
Answer:

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
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 7

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

Question 9.
J(2, – 1), K(4, – 5), L(3, 1); x = – 1
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 9

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

Question 11.
J(2, 4), K(- 4, – 2), L(- 1, 0); y = 1
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 11

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

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

Question 13.
y = x
Big Ideas Math Answers Geometry Chapter 4 Transformations 38
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 13

Question 14.
y = x
Big Ideas Math Answers Geometry Chapter 4 Transformations 39
Answer:

Question 15.
y = -x
Big Ideas Math Answers Geometry Chapter 4 Transformations 40
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 15

Question 16.
y = -x
Big Ideas Math Answers Geometry Chapter 4 Transformations 41
Answer:

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
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 17

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

Question 19.
Translation: (x, y) → (x, y + 4)
Reflection: in the line x = 3
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 19

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

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

Question 21.
Big Ideas Math Answers Geometry Chapter 4 Transformations 42
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 21

Question 22.
Big Ideas Math Answers Geometry Chapter 4 Transformations 43
Answer:

Question 23.
Big Ideas Math Answers Geometry Chapter 4 Transformations 44
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 23

Question 24.
Big Ideas Math Answers Geometry Chapter 4 Transformations 45
Answer:

Question 25.
USING STRUCTURE
Identify the line symmetry (if any) of each word.
a. LOOK
b. MOM
c. OX
d. DAD
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 25

Question 26.
ERROR ANALYSIS
Describe and correct the error in describing the transformation.
Big Ideas Math Answers Geometry Chapter 4 Transformations 46
Answer:

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?
Big Ideas Math Answers Geometry Chapter 4 Transformations 47
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 27

Question 28.
ATTENDING TO PRECISION
Use the numbers and symbols to create the glide reflection resulting in the image shown.
Big Ideas Math Answers Geometry Chapter 4 Transformations 48
Answer:

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)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 29

Question 30.
A(4, – 5), B(12, 3)
Answer:

Question 31.
A(- 8, 4), B(- 1, 3)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 31

Question 32.
A(- 1, 7), B(5, – 4)
Answer:

Question 33.
MATHEMATICAL CONNECTIONS
The line y = 3x + 2 is reflected in the line y = – 1. What is the equation of the image?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 33

Question 34.
HOW DO YOU SEE IT?
Use Figure A.
Big Ideas Math Answers Geometry Chapter 4 Transformations 49

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

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

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

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

Question 35.
CONSTRUCTION
Follow these steps to construct a reflection of △ ABC in line m. Use a compass and straightedge.
Big Ideas Math Answers Geometry Chapter 4 Transformations 50
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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 35

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

Question 37.
MATHEMATICAL CONNECTIONS
Reflect △MNQ in the line y = -2x.
Big Ideas Math Answers Geometry Chapter 4 Transformations 51
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 37

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.
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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 39

Maintaining Mathematical Proficiency

Use the diagram to Íind the angle measure.

Big Ideas Math Answers Geometry Chapter 4 Transformations 52

Question 40.
m∠AOC
Answer:

Question 41.
m∠AOD
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 41

Question 42.
m∠BOE
Answer:

Question 43.
m∠AOE
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 43

Question 44.
m∠COD
Answer:

Question 45.
m∠EOD
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 45

Question 46.
m∠COE
Answer:

Question 47.
m∠AOB
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 47

Question 48.
m∠COB
Answer:

Question 49.
m∠BOD
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.2 Question 49

4.3 Rotations

Exploration 1

Rotating a Triangle in a Coordinate Plane

Work with a partner:

Big Ideas Math Geometry Answer Key Chapter 4 Transformations 53

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

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

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

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

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.
Answer:

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’?
Answer:

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

Exploration 3

Rotating a Triangle in a Coordinate Plane

Work with a partner.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 54

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.
Answer:

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’?
Answer:

Communicate Your Answer

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

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?
Answer:

Lesson 4.3 Rotations

Monitoring Progress

Question 1.
Trace ∆DEF and point P. Then draw a 50° rotation of ∆DEF about point P.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 55
Answer:

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.
Answer:

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.
Answer:

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.
Answer:

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)
Rotation: 90° about the origin
Answer:

Question 6.
Graph ∆TUV with vertices T(1, 2), U(3. 5), and V(6, 3) and its image after the composition.
Rotation: 180° about the origin
Reflection: in the x-axis
Answer:

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

Question 7.
rhombus
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 56
Answer:

Question 8.
octagon
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 57
Answer:

Question 9.
right triangle
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 58
Answer:

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 _________ .
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 1

Question 2.
DIFFERENT WORDS, SAME QUESTION
Which is different? Find “both” answers.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 59
What are the coordinates of the vertices of the image after a 90° counterclockwise rotation about the origin?
Answer:

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

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

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

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°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 60
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 3

Question 4.
80°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 61
Answer:

Question 5.
150°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 62
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 5

Question 6.
130°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 63
Answer:

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°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 64
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 7

Question 8.
180°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 65
Answer:

Question 9.
180°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 66
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 9

Question 10.
270°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 67
Answer:

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)
Rotation: 90° about the origin
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 11

Question 12.
Rotation: 180° about the origin
Translation: (x, y) → (x – 1, y + 1)
Answer:

Question 13.
Rotation: 270° about the origin
Reflection: in the y-axis
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 13

Question 14.
Reflection: in the line y = x
Rotation: 180° about the origin
Answer:

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.
Rotation: 90° about the origin
Translation: (x, y) → (x – 3, y + 2)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 15

Question 16.
Reflection: in the x-axis
Rotation: 270° about the origin
Answer:

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

Question 17.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 68
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 17

Question 18.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 69
Answer:

Question 19.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 70
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 19

Question 20.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 71
Answer:

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.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 72
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 21

Question 22.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 73
Answer:

Question 23.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 74
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 23

Question 24.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 75
Answer:

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.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 76
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 25

Question 26.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 77
Answer:

Question 27.
CONSTRUCTION
Follow these Steps to construct a rotation of ∆ABC by angle D around a point O. Use a compass and straightedge.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 78
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’.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 27

Question 28.
REASONING
You enter the revolving door at a hotel.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 79
a. You rotate the door 180°. what does this mean in the context of the situation? Explain.
Answer:

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

Question 29.
MATHEMATICAL CONNECTIONS
Use the graph of Y = 2X – 3.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 80
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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 29

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.
Answer:

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?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 31

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

Question 33.
ANALYZING RELATIONSHIPS
Is it possible for a figure to have 180° rotational symmetry hut not 90° rotational symmetry? Explain your reasoning.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 33

Question 34.
THOUGHT PROVOKING
Can rotations of 90°, 180°, 270°, and 360° be written as the composition of two reflections? Justify your answer.
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.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 81
a.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 82

b.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 83
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 35

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.
Answer:

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).
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 37

Question 38.
HOW DO YOU SEE IT?
You are finishing the puzzle. The remaining two pieces both have rotational symmetry.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 84
a. Describe the rotational symmetry of Piece 1 and of Piece 2.
Answer:

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

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.
Answer:

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.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 85
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 Question 39

Maintaining Mathematical Proficiency

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

Question 40.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 86
Answer:

Question 41.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 87
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.3 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)
Answer:

Question 2.
(x, y) → (x – 1, y – 5)
Answer:

Question 3.
(x, y) → (x + 3, y + 6)
Answer:

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
Answer:

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

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

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

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
Answer:

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

Determine the number of lines of symmetry for the figure.

Question 10.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 88
Answer:

Question 11.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 89
Answer:

Question 12.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 90
Answer:

Question 13.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 91
Answer:

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

Question 14.
90°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 92
Answer:

Question 15.
270°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 93
Answer:

Question 16.
180°
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 94
Answer:

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)
Rotation: 180° about the origin
Answer:

Question 18.
Rotation: 90° about the origin
Reflection: in the y-axis
Answer:

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.
Big Ideas Math Geometry Answer Key Chapter 4 Transformations 95
Answer:

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.

Big Ideas Math Geometry Solutions Chapter 4 Transformations 96

a. Draw an line Big Ideas Math Geometry Solutions Chapter 4 Transformations 97. Reflect ∆ ABC in Big Ideas Math Geometry Solutions Chapter 4 Transformations 97 to form ∆A’B’C’.
Answer:

b. Draw a line parallel to Big Ideas Math Geometry Solutions Chapter 4 Transformations 97. Reflect ∆A’B’C’ in the new line to form ∆A”B”C”.
Answer:

c. Draw the line through point A that is perpendicular to Big Ideas Math Geometry Solutions Chapter 4 Transformations 97. What do you notice?
Answer:

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

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

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.
Answer:

Exploration 2

Reflections in Intersecting Lines

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

Big Ideas Math Geometry Solutions Chapter 4 Transformations 99

a. Draw an line Big Ideas Math Geometry Solutions Chapter 4 Transformations 97. Reflect ∆ABC in Big Ideas Math Geometry Solutions Chapter 4 Transformations 97 to form ∆A’B’C’.
Answer:

b. Draw any line Big Ideas Math Geometry Solutions Chapter 4 Transformations 98 so that angle EDF is less than or equal to 90°. Reflect ∆A’B’C’ in Big Ideas Math Geometry Solutions Chapter 4 Transformations 98 to form ∆A”B”C”.
Answer:

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

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

Communicate your Answer

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

Question 4.
Point Q is reflected in two parallel lines, Big Ideas Math Geometry Solutions Chapter 4 Transformations 100 and Big Ideas Math Geometry Solutions Chapter 4 Transformations 101. to form Q’ and The distance from Big Ideas Math Geometry Solutions Chapter 4 Transformations 100 to Big Ideas Math Geometry Solutions Chapter 4 Transformations 101 is 3.2 inches. What is the distance QQ”?
Answer:

Lesson 4.4 Congruence and Transformations

Monitoring Progress

Question 1.
Identify any congruent figures in the coordinate plane. Explain.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 102
Answer:

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

Question 3.
Describe a congruence transformation that maps △JKL to △MNP.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 103
Answer:

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

Big Ideas Math Geometry Solutions Chapter 4 Transformations 104

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.
Answer:

Question 5.
What is the relationship between \(\overline{P P’}\) and line k? Explain.
Answer:

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

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.
Answer:

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?
Big Ideas Math Geometry Solutions Chapter 4 Transformations 105
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 1

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

Monitoring Progress and Modeling with Mathematics

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

Question 3.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 106
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 3

Question 4.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 107
Answer:

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

Question 5.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 108
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 5

Question 6.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 109
Answer:

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)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 7

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)
Answer:

Question 9.
J(1, 1), K(3, 2), L(4, 1) and M(6, 1), N(5, 2), P(2, 1)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 9

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)
Answer:

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

Big Ideas Math Geometry Solutions Chapter 4 Transformations 110

Question 11.
A translation maps ∆ABC onto which triangle?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 11

Question 12.
Which lines are perpendicular to \(\overline{A A”}\)?
Answer:

Question 13.
If the distance between k and m is 2.6 inches. what is the length of \(\overline{C C”}\)?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 13

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

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

Question 15.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 111
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 15

Question 16.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 112
Answer:

Question 17.
ERROR ANALYSIS
Describe and correct the error in describing the congruence transformation.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 113
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 17

Question 18.
ERROR ANALYSIS
Describe and correct the error in using the Reflections in Intersecting Lines Theorem
Big Ideas Math Geometry Solutions Chapter 4 Transformations 114
Answer:

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’.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 19

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

Question 21.
The rotation (x, y) → (- x, – y) maps C to C’.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 21

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

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?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 23

Question 24.
DRAWING CONCLUSIONS
The pattern shown is called a tessellation.
Big Ideas Math Geometry Solutions Chapter 4 Transformations 115
a. What transformations did the artist use when creating this tessellation?
Answer:

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

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 25

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.
Answer:

Question 27.
The composition of two reflections results in the same image as a rotation.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 27

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

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 29

Question 30.
HOW DO YOU SEE IT?
What type of congruence transformation can be used to verify each statement about the stained glass window?
Big Ideas Math Geometry Solutions Chapter 4 Transformations 116
a. Triangle 5 is congruent to Triangle 8.
Answer:

b. Triangle 1 is congruent to Triangle 4.
Answer:

c. Triangle 2 is congruent to Triangle 7.
Answer:

d. Pen1aon 3 is congruent to Pentagon 6.
Answer:

Question 31.
PROVING A THEOREM
Prove the Reflections in Parallel Lines Theorem (Theorem 4.2).
Big Ideas Math Geometry Solutions Chapter 4 Transformations 117
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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 31.1
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 31.2

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.
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 33

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 ?
Big Ideas Math Geometry Solutions Chapter 4 Transformations 118
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.
Answer:

Question 35.
Translation: ∆ABC → ∆A”B”C”
Big Ideas Math Geometry Solutions Chapter 4 Transformations 119
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 35

Question 36.
Rotation about P: ∆XYZ → ∆X”Y”Z”
Big Ideas Math Geometry Solutions Chapter 4 Transformations 120
Answer:

Maintaining Mathematical Proficiency

Solve the equation. Check your solution.

Question 37.
5x + 16 = – 3x
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 37

Question 38.
12 + 6m = 2m
Answer:

Question 39.
4b + 8 = 6b – 4
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 39

Question 40.
7w – 9 = 13 – 4w
Answer:

Question 41.
7(2n + 11) = 4n
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 41

Question 42.
-2(8 – y) = – 6y
Answer:

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?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.4 Question 43

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’.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 121
Answer:

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.
Answer:

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?
Answer:

Exploration 2

Dilating Lines in a Coordinate Plane

Work with a partner. Use dynamic geometry software to draw Big Ideas Math Answer Key Geometry Chapter 4 Transformations 122 that passes through the origin and Big Ideas Math Answer Key Geometry Chapter 4 Transformations 123 that does not pass through the origin.

Big Ideas Math Answer Key Geometry Chapter 4 Transformations 124

a. Dilate Big Ideas Math Answer Key Geometry Chapter 4 Transformations 122 using a scale factor of 3 and a center of dilation at the origin. Describe the image.
Answer:

b. Dilate Big Ideas Math Answer Key Geometry Chapter 4 Transformations 123 using a scale factor of 3 and a center of dilation at the origin. Describe the image.
Answer:

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

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?
Answer:

Communicate Your Answer

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

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

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.
Answer:

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
Answer:

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

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.
Answer:

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
Answer:

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.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 125
Answer:

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 _________.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 1

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
Answer:

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.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 126
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 3

Question 4.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 127
Answer:

Question 5.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 128
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 5

Question 6.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 129
Answer:

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.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 130

Question 7.
Center C, k = 2
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 7

Question 8.
Center P, k = 3
Answer:

Question 9.
Center M, k = \(\frac{1}{2}\)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 9

Question 10.
Center C. k = 25%
Answer:

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.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 131
Question 11.
Center C, k = 3
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 11

Question 12.
Center P, k = \(\frac{1}{3}\)
Answer:

Question 13.
Center P, k = 0.25
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 13

Question 14.
Center C, k = 75%
Answer:

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
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 15

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

Question 17.
T(9, – 3), U(6, 0), V(3, 9), W(0. 0); k = \(\frac{2}{3}\)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 17

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

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
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 19

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

Question 21.
R(- 7, – 1), S(2, 5), T(- 2, – 3), U(- 3,- 3); k = – 4
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 21

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

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

Question 23.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 132
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 23

Question 24.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 133
Answer:

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.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 134
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 25

Question 26.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 135
Answer:

Question 27.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 136
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 27

Question 28.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 137
Answer:

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?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 29

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?
Answer:

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×
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 138
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 31

Question 32.
ladybug
Magnification: 10×
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 139
Answer:

Question 33.
dragonfly
Magnification: 20×
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 140
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 33

Question 34.
carpenter ant
Magnification: 15×
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 141
Answer:

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.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 142
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 35

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.
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 37

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.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 143
Answer:

Question 39.
MATHEMATICAL CONNECTIONS
The larger triangle is a dilation of the smaller triangle. Find the values of x and y.
Big Ideas Math Answer Key Geometry Chapter 4 Transformations 144
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 39

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

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
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 41

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

Question 43.
Center of dilation: outside the figure; k = 120%
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 43

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

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 Big Ideas Math Answer Key Geometry Chapter 4 Transformations 145
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 45

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 \(\overline{A B}\)?
Answer:

b. What do you notice about Big Ideas Math Answer Key Geometry Chapter 4 Transformations 146?
Answer:

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?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 47

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.
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 49.1
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 49.2
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 49.3
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 49.4

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.
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 51

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)
Answer:

Question 53.
(x, y) → (x – 1, y + 3)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 53

Question 54.
(x, y) → (x + 3, y – 1)
Answer:

Question 55.
(x, y) → (x – 2, y)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 55

Question 56.
(x, y) → (x + 1, y – 2)
Answer:

Question 57.
(x, y) → (x – 3y + 1)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.5 Question 57

4.6 Similarity and Transformations

Exploration 1

Dilations and similarity

Work with a partner.

Big Ideas Math Geometry Answers Chapter 4 Transformations 147

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

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.
Answer:

Exploration 2

Rigid Motions and Similarity

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

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.
Answer:

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

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

Communicate 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.
Answer:

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

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.
Rotation: 90° about the origin
Dilation: (x, y) → \(\left(\frac{1}{2} x, \frac{1}{2} y\right)\)
Answer:

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)
Answer:

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

Question 4.
Describe a similarity transformation that maps quadrilateral DEFG to quadrilateral STUV.
Big Ideas Math Geometry Answers Chapter 4 Transformations 148
Answer:

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 149
Answer:

Exercise 4.6 Similarity and Transformations

Vocabulary and Core Concept Check

Question 1.
VOCABULARY
What is the difference between similar figures and congruent figures?
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 1

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

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)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 3

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

Question 5.
Rotation: 90° about the origin
Dilation: (x, y) → (3x, 3y)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 5

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

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

Question 7.
Big Ideas Math Geometry Answers Chapter 4 Transformations 150
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 7

Question 8.
Big Ideas Math Geometry Answers Chapter 4 Transformations 151
Answer:

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)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 9

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)
Answer:

Question 11.
G(- 2, 3), H(4, 3), I(4, 0) and J(1, 0), K(6, – 2), L(1, – 2)
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 11

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)
Answer:

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 152
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 13.1
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 13.2
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 13.3

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 153
Answer:

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 154
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 15

Question 16.
ERROR ANALYSIS
Describe and correct the error in comparing the figures.
Big Ideas Math Geometry Answers Chapter 4 Transformations 155
Answer:

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.
Big Ideas Math Geometry Answers Chapter 4 Transformations 156
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 17

Question 18.
HOW DO YOU SEE IT?
Determine whether each pair of figures is similar. Explain your reasoning.
a.
Big Ideas Math Geometry Answers Chapter 4 Transformations 157
Answer:

b.
Big Ideas Math Geometry Answers Chapter 4 Transformations 158
Answer:

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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 19

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.
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.
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 21

Question 22.
REPEATED REASONING
Use the diagram.
Big Ideas Math Geometry Answers Chapter 4 Transformations 159
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.
Answer:

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

Maintaining Mathematical Proficiency

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

Question 23.
Big Ideas Math Geometry Answers Chapter 4 Transformations 160
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 23

Question 24.
Big Ideas Math Geometry Answers Chapter 4 Transformations 161
Answer:

Question 25.
Big Ideas Math Geometry Answers Chapter 4 Transformations 162
Answer:
Big Ideas Math Geometry Answers Chapter 4 Transformations 4.6 Question 25

Question 26.
Big Ideas Math Geometry Answers Chapter 4 Transformations 163
Answer:

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)
Answer:

Question 2.
(x, y) → (x – 3, y)
Answer:

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

Question 4.
(x, y) → (x + 4, y + 1)
Answer:

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)
Answer:

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

4.2 Reflections

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

Question 7.
x = 4
Big Ideas Math Geometry Answers Chapter 4 Transformations 164
Answer:

Question 8.
y = 3
Big Ideas Math Geometry Answers Chapter 4 Transformations 165
Answer:

Question 9.
How many lines of symmetry does the figure have?
Big Ideas Math Geometry Answers Chapter 4 Transformations 166
Answer:

4.3 Rotations

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

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

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.
Answer:

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

Question 13.
Big Ideas Math Geometry Answers Chapter 4 Transformations 167
Answer:

Question 14.
Big Ideas Math Geometry Answers Chapter 4 Transformations 168
Answer:

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)
Answer:

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

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

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 = \(\frac{1}{2}\)
Answer:

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

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.
Answer:

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)
Answer:

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

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

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)
Answer:

Question 2.
(x, y) → (x + 2, y – 2)
Answer:

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°
Answer:

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

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)
Answer:

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

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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 169
Answer:

Question 8.
Big Ideas Math Answers Geometry Chapter 4 Transformations 170
Answer:

Question 9.
Big Ideas Math Answers Geometry Chapter 4 Transformations 171
Answer:

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).
Answer:

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.
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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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 172
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Question 13.
There is one slice of a large pizza and one slice of a small pizza in the box.
Big Ideas Math Answers Geometry Chapter 4 Transformations 173
a. Describe a similarity transformation that maps pizza slice ABC to pizza slice DEF.
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b. What is one possible scale factor for a medium slice of pizza? Explain your reasoning.
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Question 14.
The original photograph shown is 4 inches by 6 inches.
Big Ideas Math Answers Geometry Chapter 4 Transformations 174
a. What transformations can you use to produce the new photograph?
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b. You dilate the original photograph b a scale factor of \(\frac{1}{2}\). What are the dimensions of the new photograph?
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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.
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Transformations Cumulative Assessment

Question 1.
Which composition 0f transformations maps ∆ABC to ∆DEF?
Big Ideas Math Answers Geometry Chapter 4 Transformations 175
(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)
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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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 176
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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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 177
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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.
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Question 5.
The graph shows quadrilateral WXYZ and quadrilateral ABCD.
Big Ideas Math Answers Geometry Chapter 4 Transformations 178
a. Write a composition of transformations that maps quadrilateral WXYZ to
quadrilateral ABCD.
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b. Are the quadrilaterals congruent? Explain your reasoning.
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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
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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.
Big Ideas Math Answers Geometry Chapter 4 Transformations 179
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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.
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b. A reflection in the x-axis maps quadrilateral ABCD onto itself
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c. A rotation of 90° about the origin maps quadrilateral ABCD onto itself.
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d. A rotation of 180° about the origin maps quadrilateral ABCD onto itself.
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