# Go Math Grade 8 Answer Key Chapter 9 Transformations and Congruence

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## Go Math Grade 8 Answer Key Chapter 9 Transformations and Congruence

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Lesson 1: Properties of Translations

Lesson 2: Properties of Reflections

Lesson 3: Algebraic Representations of Transformations

Lesson 4: Congruent Figures

Model Quiz

Mixed Review

### Guided Practice – Properties of Translations – Page No. 282

Question 1.
Vocabulary A __________________is a change in the position, size, or shape of a figure.
____________

transformation

Explanation:
A transformation is a change in the position, size, or shape of a figure.

Question 2.
Vocabulary When you perform a transformation of a figure on the coordinate plane, the input of the transformation is called the ________________, and the output of the transformation is called the_________________ .
Type below:
____________

pre-image
image

Explanation:
When you perform a transformation of a figure on the coordinate plane, the input of the transformation is called the pre-image, and the output of the transformation is called the image.

Question 3.
Joni translates a right triangle 2 units down and 4 units to the right. How does the orientation of the image of the triangle compare with the orientation of the preimage?
Orientation is: _______

Orientation is: Same

Explanation:
Since translation does not change the shape and size of a geometric figure, the two triangles are identical in shape and size, so they are congruent and the orientation is the same

Question 4.
Rashid drew rectangle PQRS on a coordinate plane. He then translated the rectangle 3 units up and 3 units to the left and labeled the image P ‘Q ‘R ‘S ‘. How do rectangle PQRS and rectangle P ‘Q ‘R ‘S ‘ compare?
They are: _______

congruent

Explanation:
Since translation does not change the shape and size of a geometric figure, the two rectangles are identical in shape and size, so they are congruent.

Question 5.
The figure shows trapezoid WXYZ. Graph the image of the trapezoid after a translation of 4 units up and 2 units to the left.

Type below:
____________

After translation:
W'(-4, 3)
X'(2, 3)
Y'(1, 1)
Z'(-3, 1)

ESSENTIAL QUESTION CHECK-IN

Question 6.
What are the properties of translations?
Type below:
____________

-> a translation is a geometric transformation that moves every point of a figure or space by the same amount in a given direction.
-> So the figures are identical and are congruent.

### 9.1 Independent Practice – Properties of Translations – Page No. 283

Question 7.
The figure shows triangle DEF.

a. Graph the image of the triangle after the translation that maps point D to point D ‘.
Type below:
____________

2 left, and 4 down

Question 7.
b. How would you describe the translation?
Type below:
____________

It has the same size, shape. and orientation, but a different location

Question 7.
c. How does the image of triangle DEF compare with the preimage?
____________

congruent

Question 8.
a. Graph quadrilateral KLMN with vertices K(-3, 2), L(2, 2), M(0, -3), and N(-4, 0) on the coordinate grid.

Type below:
____________

Question 8.
b. On the same coordinate grid, graph the image of quadrilateral KLMN after a translation of 3 units to the right and 4 units up.
Type below:
____________

Question 8.
c. Which side of the image is congruent to side $$\overline { LM }$$?
___________
Name three other pairs of congruent sides.
___________
Type below:
____________

Line LM is congruent to Line L!M!
Line KL is equal to K’L’
Line MN is equal to M’N’
Line KN is equal to K’N’

Draw the image of the figure after each translation.

Question 9.
4 units left and 2 units down

Type below:
____________

After translation
P'(-3, 1)
Q'(0, 2)
R'(0, -1)
S'(-3, -3)

Question 10.
5 units right and 3 units up

Type below:
____________

After translation
A'(0, 4)
B'(3, 5)
C'(3, 1)
D'(0, 0)

### Properties of Translations – Page No. 284

Question 11.
The figure shows the ascent of a hot air balloon. How would you describe the translation?

Type below:
____________

4 units along positive X and 5 units along positive Y

Explanation:
Initial coordinate of balloon = ( -2 , -4)
Final coordinates of the balloon = (2,1)
Translation along x axis = 2 – (-2)
= 4 units along positive x direction
Translation along y axis = 1-(-4)
= 5 units along the positive y direction

Question 12.
Critical Thinking Is it possible that the orientation of a figure could change after it is translated? Explain.
_________

No, it is not possible to change the orientation just by translation. As translation means, a transformation in which a figure is moved to another location without any change in size or orientation.

FOCUS ON HIGHER ORDER THINKING

Question 13.
a. Multistep Graph triangle XYZ with vertices X(-2, -5), Y(2, -2), and Z(4, -4) on the coordinate grid.

Question 13.
b. On the same coordinate grid, graph and label triangle X’Y’Z’, the image of triangle XYZ after a translation of 3 units to the left and 6 units up.

Question 13.
c. Now graph and label triangle X”Y”Z”, the image of triangle X’Y’Z’ after a translation of 1 unit to the left and 2 units down.
Type below:
____________

Question 13.
d. Analyze Relationships How would you describe the translation that maps triangle XYZ onto triangle X”Y”Z”?
Type below:
____________

Triangle XYZ has translated 4 units up and 4 units to the left

Question 14.
Critical Thinking The figure shows rectangle P’Q’R’S’, the image of rectangle PQRS after a translation of 5 units to the right and 7 units up. Graph and label the preimage PQRS.

Type below:
____________

Question 15.
Communicate Mathematical Ideas Explain why the image of a figure after a translation is congruent to its preimage.
Type below:
____________

A translation is a geometric transformation that moves every point of a figure or space by the same amount in a given direction. So the 2 figures are identical and the translated figure is congruent to its pre-image.

### Guided Practice – Properties of Reflections – Page No. 288

Question 1.
Vocabulary A reflection is a transformation that flips a figure across a line called the __________ .
____________

Reflection Axis

Explanation:
A reflection is a transformation that flips a figure across a line called the Reflection Axis.

Question 2.
The figure shows trapezoid ABCD.

a. Graph the image of the trapezoid after a reflection across the x-axis. Label the vertices of the image.
Type below:
____________

A'(-3, -4)
B'(1, -4)
C'(3, -1)
D'(-3, -1)

Question 2.
b. How do trapezoid ABCD and trapezoid A’B’C’D’ compare?
____________

congruent

Explanation:
trapezoid ABCD and trapezoid A’B’C’D’ are congruent

Question 2.
c. What If? Suppose you reflected trapezoid ABCD across the y-axis. How would the orientation of the image of the trapezoid compare with the orientation of the preimage?
Type below:
____________

The orientation would be reversed horizontally.

ESSENTIAL QUESTION CHECK-IN

Question 3.
What are the properties of reflections?
Type below:
____________

properties of reflections
-> the size stays the same
-> the shape stays the same
-> the orientation does NOT stay the same

### 9.2 Independent Practice – Properties of Reflections – Page No. 289

The graph shows four right triangles. Use the graph for Exercises 4-7.

Question 4.
Which two triangles are reflections of each other across the x-axis?
Type below:
____________

Triangles A and C are the reflections of each other across the x-axis.

Question 5.
For which two triangles is the line of reflection the y-axis?
Type below:
____________

For triangle C & D the line of reflection is y-axis.

Question 6.
Which triangle is a translation of triangle C? How would you describe the translation?
Type below:
____________

Triangle B is the translation of triangle C.
Lets take any one point of the triangle = (-2, -6)
Lets take the corresponding side of triangle B = (4,2)
Translation across x axis = 4 -(-2) = 6 units
Translation across y axis = 2 -(-6) = 8 units

Question 7.
Which triangles are congruent? How do you know?
Type below:
____________

All the 4 triangles A, B, C, D are congruent.
The length of base and height of all the four triangles are 3 units, 4 units respectively.

Explanation:
All the 4 triangles A, B, C, D are congruent.
If base and height are equal then the hypotenuse should also be equal. Thus all three sides of the triangles A,B,C,D are equal. Thus these triangles are congruent,
The length of base and height of all the four triangles are 3 units, 4 units respectively.

Question 8.
a. Graph quadrilateral WXYZ with vertices W(-2, -2), X(3, 1), Y(5, -1), and Z(4, -6) on the coordinate grid.
Type below:
____________

Question 8.
b. On the same coordinate grid, graph quadrilateral W’X’Y’Z’, the image of quadrilateral WXYZ after a reflection across the x-axis.
Type below:
____________

Question 8.
c. Which side of the image is congruent to side $$\overline { YZ }$$?
_______________
Name three other pairs of congruent sides.
_______________
Type below:
____________

Line YZ = Line Y’Z’
Line WX = Line W’X’
Line XY = Line X’Y’
Line WZ = Line W’Z’

Question 8.
d. Which angle of the image is congruent to ∠X?
_______________
Name three other pairs of congruent angles.
_______________
Type below:
____________

Angle X’
Angle W and Angle W’
Angle Y and Angle Y’
Angle Z and Angle Z’

### Properties of Reflections – Page No. 290

Question 9.
Critical Thinking Is it possible that the image of a point after a reflection could be the same point as the preimage? Explain.
________

Yes

Explanation:
It is possible that the image of a point after a reflection could be the same point as the preimage

FOCUS ON HIGHER ORDER THINKING

Question 10.
a. Graph the image of the figure shown after a reflection across the y-axis.

Type below:
____________

Question 10.
b. On the same coordinate grid, graph the image of the figure you drew in part a after a reflection across the x-axis.
Type below:
____________

Question 10.
c. Make a Conjecture What other sequence of transformations would produce the same final image from the original preimage? Check your answer by performing the transformations. Then make a conjecture that generalizes your findings.
Type below:
____________

The same image can be obtained by reflecting first across the x-axis and then across the y-axis.
Reflecting a figure first across the y-axis and then across the x-axis has the same outcome,. reflecting first across the x-axis and then across the y-axis.

Question 11.
a. Graph triangle DEF with vertices D(2, 6), E(5, 6), and F(5, 1) on the coordinate grid.

Question 11.
b. Next graph triangle D ′E ′F ′, the image of triangle DEF after a reflection across the y-axis.
Type below:
____________

Question 11.
c. On the same coordinate grid, graph triangle D′′ E′′ F′′, the image of triangle D ′E ′F ′ after a translation of 7 units down and 2 units to the right.
Type below:
____________

Question 11.
d. Analyze Relationships Find a different sequence of transformations that will transform triangle DEF to triangle D ′′E ′′F ′′.
Type below:
____________

Translate triangle DEF 7 units down and 2 units to the left. Then reflect the image across the y-axis.

### Guided Practice – Properties of Reflections – Page No. 294

Question 1.
Vocabulary A rotation is a transformation that turns a figure around a given _____ called the center of rotation.
____________

point

Explanation:
A rotation is a transformation that turns a figure around a given point called the center of rotation.

Siobhan rotates a right triangle 90° counterclockwise about the origin.

Question 2.
How does the orientation of the image of the triangle compare with the orientation of the preimage?
Type below:
____________

Each leg in the preimage is perpendicular to its corresponding leg in the image.

Question 3.
Is the image of the triangle congruent to the preimage?
______

Yes

Explanation:
The image of the triangle is congruent to the preimage

Draw the image of the figure after the given rotation about the origin.

Question 4.
90° counterclockwise

Type below:
____________

Question 5.
180°

Type below:
____________

After 180° rotation
A'(-2, -3)
B'(-4, -1)
C'(-2, 0)
D'(0, -1)

ESSENTIAL QUESTION CHECK-IN

Question 6.
What are the properties of rotations?
Type below:
____________

Rotations preserve size and shape but change orientation.

### 9.3 Independent Practice – Properties of Reflections – Page No. 295

Question 7.
The figure shows triangle ABC and a rotation of the triangle about the origin.

a. How would you describe the rotation?
____________

ABC was rotated 90º counterclockwise about the origin

Question 7.
b. What are the coordinates of the image?
Type below:
____________

A'(3, 1)
B'(2, 3)
C'(-1, 4)

Question 8.
The graph shows a figure and its image after a transformation.

a. How would you describe this as a rotation?
____________

The figure was rotated 180º about the origin.

Question 8.
b. Can you describe this as a transformation other than a rotation? Explain.
____________

Yes

Explanation:
This can also be described as a reflection across the y-axis.

Question 9.
What type of rotation will preserve the orientation of the H-shaped figure in the grid?

____________

A 180º rotation about the origin will preserve the orientation of the H-shaped figure in the grid.

Question 10.
A point with coordinates (-2, -3) is rotated 90° clockwise about the origin. What are the coordinates of its image?
(_______ , _______)

(-3, 2)

Explanation:
The new coordinates are (-3, 2)

Complete the table with rotations of 180° or 90°. Include the direction of rotation for rotations of 90°.

Question 11.

Type below:
____________

### Properties of Reflections – Page No. 296

Draw the image of the figure after the given rotation about the origin.

Question 14.
180°

Type below:
____________

After 180°
A'(4, 0)
B'(2, -1)
C'(0, 0)
D'(2, 1)

Question 15.
270° counterclockwise

Type below:
____________

After 270º counterclockwise rotation
A'(1, 2)
B'(2, -1)
C'(4, 2)

Question 16.
Is there a rotation for which the orientation of the image is always the same as that of the preimage? If so, what?
______

Yes

Explanation:
A 360º rotation will always be the same as the original image

FOCUS ON HIGHER ORDER THINKING

Question 17.
Problem Solving Lucas is playing a game where he has to rotate a figure for it to fit in an open space. Every time he clicks a button, the figure rotates 90 degrees clockwise. How many times does he need to click the button so that each figure returns to its original orientation?
Figure A ____________
Figure B ____________
Figure C ____________

Figure A: _________ time(s)
Figure B: _________ time(s)
Figure C: _________ time(s)

Figure A: 2 time(s)
Figure B: 1 time(s)
Figure C: 4 time(s)

Explanation:

Question 18.
Make a Conjecture Triangle ABC is reflected across the y-axis to form the image A′B′C′. Triangle A′B′C′ is then reflected across the x-axis to form the image A″B″C″. What type of rotation can be used to describe the relationship between triangle A″B″C″ and triangle ABC?
Type below:
____________

Triangle A’B’C’ is a 90º rotation of triangle ABC
Triangle A”B”C” is a 90º rotation of triangle A’B’C’
Therefore, Triangle A”B”C” is a 180º rotation of triangle ABC

Question 19.
Communicate Mathematical Ideas Point A is on the y-axis. Describe all possible locations of image A′ for rotations of 90°, 180°, and 270°. Include the origin as a possible location for A.
Type below:
____________

If Point A is on the y-axis, Point A’ will be on the x-axis for 190° and 270° rotations and on the y-axis for 180° rotation
If point A is at the origin,
A’ is at the origin for any rotation about the origin.

### Guided Practice – Algebraic Representations of Transformations – Page No. 300

Question 1.
Triangle XYZ has vertices X(-3, -2), Y(-1, 0), and Z(1, -6). Find the vertices of triangle X′Y′Z′ after a translation of 6 units to the right. Then graph the triangle and its image.

Type below:
____________

After a translation of 6 units to the right:
X'(3, -2)
Y'(5, 0)
Z'(7, -6)

Question 2.
Describe what happens to the x- and y-coordinates after a point is reflected across the x-axis.
Type below:
____________

The x-coordinate remains the same, while the sign of the y-coordinate changes

Question 3.
Use the rule (x, y) → (y, -x) to graph the image of the triangle at right. Then describe the transformation.

Type below:
____________

The triangle is rotated 90º clockwise about the origin

ESSENTIAL QUESTION CHECK-IN

Question 4.
How do the x- and y-coordinates change when a figure is translated right a units and down b units?
Type below:
____________

The x-coordinates increase by a, and the y-coordinates decrease by b

### 9.4 Independent Practice – Algebraic Representations of Transformations – Page No. 301

Write an algebraic rule to describe each transformation.Then describe the transformation.

Question 5.

Type below:
____________

algebraic rule
(x, y) -> (x-2, y-5)
transformation
translation of 2 units to the left and 5 units down
new coordinates
M'(-4, -2)
N'(-2, -2)
O'(-1, -4)
P'(-4, -4)

Question 6.

Type below:
____________

algebraic rule
(x, y) -> (-x, -y)
transformation
rotation of 180º
new coordinates
A'(0, 0)
B'(0, -3)
C'(2, -3)
D'(2, 0)

Question 7.
Triangle XYZ has vertices X(6, -2.3), Y(7.5, 5), and Z(8, 4). When translated, X′ has coordinates (2.8, -1.3). Write a rule to describe this transformation. Then find the coordinates of Y′ and Z′.
Type below:
____________

algebraic rule
(x, y) -> (x-3.2, y+1)
new coordinates
Y'(4.3, 6)
Z'(4.8, 5)

Question 8.
Point L has coordinates (3, -5). The coordinates of point L′ after a reflection are (-3, -5). Without graphing, tell which axis point L was reflected across. Explain your answer.
____________

Point L was reflected on the y-axis.
When you reflect a point across the y-axis, the sign of the x-coordinate changes, and the sign of the y-coordinate remains the same

Question 9.
Use the rule (x, y) → (x – 2, y – 4) to graph the image of the rectangle. Then describe the transformation.

Type below:
____________

The rectangle is translated 2 units to the left and 4 units down

Question 10.
Parallelogram ABCD has vertices A(−2, −5$$\frac{1}{2}$$), B(−4, −5$$\frac{1}{2}$$),C(-3, -2), and D(-1, -2). Find the vertices of parallelogram A′B′C′D′ after a translation of 2 $$\frac{1}{2}$$ units down.
Type below:
__________

after a translation of 2 $$\frac{1}{2}$$ units
A'(-2, -8)
B'(-4, -8)
C'(-3, -4 $$\frac{1}{2}$$)
D'(-1, -4 $$\frac{1}{2}$$)

### Algebraic Representations of Transformations – Page No. 302

Question 11.
Alexandra drew the logo shown on half-inch graph paper. Write a rule that describes the translation Alexandra used to create the shadow on the letter A.

Type below:
__________

(x,y) –> (x+1,y-0.5)
(x+1,y-0.5) –> (x+0.5,y-0.25)

Explanation:
translation in units
(x,y) –> (x+1,y-0.5)
This step converts translation rule in units to translation rule in inches. (Divide by 2 since graph paper is half inch paper.
(x+1,y-0.5) –> (x+0.5,y-0.25)

Question 12.
Kite KLMN has vertices at K(1, 3), L(2, 4), M(3, 3), and N(2, 0). After the kite is rotated, K′ has coordinates (-3, 1). Describe the rotation, and include a rule in your description. Then find the coordinates of L′, M′, and N′.
Type below:
__________

rotation
90 counterclockwise
rule
(x, y) -> (-y, x)
new coordinates
L'(-4, 2)
M'(-3, 3)
N'(0, 2)

FOCUS ON HIGHER ORDER THINKING

Question 13.
Make a Conjecture Graph the triangle with vertices (-3, 4), (3, 4), and (-5, -5). Use the transformation (y, x) to graph its image.
a. Which vertex of the image has the same coordinates as a vertex of the original figure? Explain why this is true.

Type below:
__________

(-5, 5) has the same coordinates

Question 13.
b. What is the equation of a line through the origin and this point?
Type below:
__________

x and y are equal so switching x and y has no effect on the coordinates

Question 13.
c. Describe the transformation of the triangle.
Type below:
__________

x and y are equal so switching x and y has no effect on the coordinates

Question 14.
Critical Thinking Mitchell says the point (0, 0) does not change when reflected across the x- or y-axis or when rotated about the origin. Do you agree with Mitchell? Explain why or why not.
_______

Yes, I agree with Mitchell

Explanation:
Reflecting across the x-axis or y-axis changes the sign of the y or x coordinate 0 cannot change signs.
Rotating about the origin does not change the origin (0, 0)

Question 15.
Analyze Relationships Triangle ABC with vertices A(-2, -2), B(-3, 1), and C(1, 1) is translated by (x, y) → (x – 1, y + 3). Then the image, triangle A′B′C′, is translated by (x, y) → (x + 4, y – 1), resulting in A″B″C″.
a. Find the coordinates for the vertices of triangle A″B″C″.
Type below:
__________

A”(-2-1+4, -2+3-1) = A”(1, 0)
B”(-3-1+4, 1+3-1) = B”(0, 3)
C”(1-1+4, 1+3-1) = C”(4, 3)

Question 15.
b. Write a rule for one translation that maps triangle ABC to triangle A″B″C″.
Type below:
__________

(x, y) -> (x-1+4, y+3-1)
(x, y) -> (x+3, y+2)

### Guided Practice – Congruent Figures – Page No. 306

Question 1.
Apply the indicated series of transformations to the rectangle. Each transformation is applied to the image of the previous transformation, not the original figure. Label each image with the letter of the transformation applied.

A. Reflection across the y-axis
B. Rotation 90° clockwise around the origin
C. (x, y) → (x – 2, y)
D. Rotation 90° counterclockwise around the origin
E. (x, y) → (x – 7, y – 2)
Type below:
__________

A. After transformation
(1, 3)
(1, 4)
(4, 4)
(4, 3)
B. After transformation
(3, -1)
(4, -1)
(4, -4)
(3, -4)
C. After transformation
(1, -1)
(2, -1)
(2, -4)
(1, -4)
D. After transformation
(1, 1)
(1, 2)
(4, 2)
(4, 1)
E. After transformation
(-6, -1)
(-6, 0)
(-3, 0)
(-3, -1)

Identify a sequence of transformations that will transform figure A into figure C.

Question 2.
What transformation is used to transform figure A into figure B?
Type below:
__________

Reflection across the y-axis

Explanation:
Reflection across the y-axis is used to transform figure A into figure B

Question 3.
What transformation is used to transform figure B into figure C?
Type below:
__________

Translation 3 units right and 4 units down

Explanation:
Translation 3 units right and 4 units down is used to transform figure B into figure C

Question 4.
What sequence of transformations is used to transform figure A into figure C? Express the transformations algebraically.
Type below:
__________

Reflection across the y-axis is used to transform figure A into figure B
Translation 3 units right and 4 units down is used to transform figure B into figure C
Algebraically:
(x, y) -> (-x, y)
(x, y) -> (x +3, y-4)

Question 5.
Vocabulary What does it mean for two figures to be congruent?
Type below:
__________

Two figures are congruent when the figures have the same size and the same shape.

ESSENTIAL QUESTION CHECK-IN

Question 6.
After a sequence of translations, reflections, and rotations, what is true about the first figure and the final figure?
Type below:
__________

After a sequence of translations, reflections, and rotations, the first and final figures have the same size and shape. (They are congruent)

### 9.5 Independent Practice – Congruent Figures – Page No. 307

For each given figure A, graph figures B and C using the given sequence of transformations. State whether figures A and C have the same or different orientation.

Question 7.

Figure B: a translation of 1 unit to the right and 3 units up
Figure C: a 90° clockwise rotation around the origin
Type below:
__________

Different orientation

Explanation:

Different orientation

Question 8.

Figure B: a reflection across the y-axis
Figure C: a 180° rotation around the origin
Type below:
__________

Different orientation

Explanation:

Different orientation

Question 9.

Figure B: a reflection across the y-axis
Figure C: a translation 2 units down
Type below:
__________

Different orientation

Explanation:

Different orientation

Question 10.

Figure B: a translation 2 units up
Figure C: a rotation of 180° around the origin
Type below:
__________

Different orientation

Explanation:

Different orientation

### Congruent Figures – Page No. 308

Question 11.
Represent Real-World Problems A city planner wanted to place the new town library at site A. The mayor thought that it would be better at site B. What transformations were applied to the building at site A to relocate the building to site B? Did the mayor change the size or orientation of the library?

Type below:
__________

From Site A to Site B: Translation 2 units right and 4 units down
The size did NOT change
The orientation changed

Question 12.
Persevere in Problem Solving Find a sequence of three transformations that can be used to obtain figure D from figure A. Graph the figures B and C that are created by the transformations.

Type below:
__________

From figure A to D:
Reflection across the x-axis (-1, -5) (-1, -6) (2, -5) (4, -6)
90º clockwise rotation (4, -1) (5, -1) (5, -4) (4, -6)
translation 6 units left (4, -1) (5, -1) (5, -4) (4, -6)

FOCUS ON HIGHER ORDER THINKING

Question 13.
Counterexamples The Commutative Properties for Addition and Multiplication state that the order of two numbers being added or multiplied does not change the sum or product. Are translations and rotations commutative? If not, give a counterexample.
________

No, Translation and rotations are not commutative

Explanation:
The point (2, 2) becomes (2, -4) when translated 2 units to the right then rotated 90 around the origin.
The point (2, 2) becomes (4, -2) when rotated 90 around the origin then translated 2 units to the right.
The above two points are not the same.

Question 14.
Multiple Representations For each representation, describe a possible sequence of transformations.
a. (x, y) → (-x – 2, y + 1)
Type below:
____________

translation 2 units right and 1 unit up
reflection across y-axis

Question 14.
b. (x, y) → (y, -x – 3)
Type below:
____________

rotation 90º clockwise around the origin
translation 3 units down

### Ready to Go On? – Model Quiz – Page No. 309

9.1–9.3 Properties of Translations, Reflections, and Rotations

Question 1.
Graph the image of triangle ABC after a translation of 6 units to the right and 4 units down. Label the vertices of the image A’, B’, and C’.

Type below:
____________

After translation:
A'(2, 1)
B'(2, -1)
C'(5, -1)

Question 2.
On the same coordinate grid, graph the image of triangle ABC after a reflection across the x-axis. Label the vertices of the image A”, B”, and C”.
Type below:
____________

After reflection:
A”(-4, -5)
B”(-4, -3)
C”(-1, -3)

Question 3.
Graph the image of HIJK after it is rotated 180° about the origin. Label the vertices of the image H’I’J’K’.

Type below:
____________

After rotation:
H'(0, -4)
I'(0, -1)
J'(2, -2)
K'(2, -3)

9.4 Algebraic Representations of Transformations

Question 4.
A triangle has vertices at (2, 3), (−2, 2), and (−3, 5). What are the coordinates of the vertices of the image after the translation (x, y) → (x + 4, y − 3)?
Type below:
____________

After translation:
(6, 0)
(2, -1)
(1, 2)

9.5 Congruent Figures

Question 5.
Vocabulary Translations, reflections, and rotations produce a figure that is _____ to the original figure.
Type below:
____________

congruent

Explanation:
Vocabulary Translations, reflections, and rotations produce a figure that is congruent to the original figure.

Question 6.
Use the coordinate grid for Exercise 3. Reflect H’I’J’K’ over the y-axis, then rotate it 180° about the origin. Label the new figure H″I″J″K″.
Type below:
____________

after reflection
H'(0, -4)
I'(0, -1)
J'(-2, -2)
K'(-2, -3)
after rotation
H”(0, 4)
I”(0, 1)
J”(2, 2)
K”(2, 3)

ESSENTIAL QUESTION

Question 7.
How can you use transformations to solve real-world problems?
Type below:
____________

Transformational properties allow the systematic movement of congruent figures while maintaining or adjusting their orientation.

### Selected Response – Mixed Review – Page No. 310

Question 1.
What would be the orientation of the figure L after a translation of 8 units to the right and 3 units up?

Options:
a. A
b. B
c. C
d. D

c. C

Explanation:
After a translation of 8 units right and 3 units up, the orientation of figure L stays the same.

Question 2.
Figure A is reflected over the y-axis and then lowered 6 units. Which sequence describes these transformations?
Options:
a. (x, y) -> (x, -y) and (x, y) -> (x, y – 6)
b. (x, y) -> (-x, y) and (x, y) -> (x, y – 6)
c. (x, y) -> (x, -y) and (x, y) -> (x – 6, y)
d. (x, y) -> (-x, y) and (x, y) -> (x – 6, y)

b. (x, y) -> (-x, y) and (x, y) -> (x, y – 6)

Explanation:
reflection over y-axis:
(x, y) -> (-x, y)
Translation 6 units down
(x, y) -> (x, y-6)

Question 3.
What quadrant would the triangle be in after a rotation of 90° counterclockwise about the origin?

Options:
a. I
b. II
c. III
d. IV

d. IV

Explanation:
After a rotation of 90° counterclockwise about the origin, the triangle will be in QIV

Question 4.
Which rational number is greater than −3 $$\frac{1}{3}$$ but less than −$$\frac{4}{5}$$?
Options:
a. −0.4
b. −$$\frac{9}{7}$$
c. −0.19
d. −$$\frac{22}{5}$$

b. −$$\frac{9}{7}$$

Question 5.
Which of the following is not true of a trapezoid that has been reflected across the x-axis?
Options:
a. The new trapezoid is the same size as the original trapezoid.
b. The new trapezoid is the same shape as the original trapezoid.
c. The new trapezoid is in the same orientation as the original trapezoid.
d. The x-coordinates of the new trapezoid are the same as the x-coordinates of the original trapezoid.

d. The x-coordinates of the new trapezoid are the same as the x-coordinates of the original trapezoid.

Explanation:
The x-coordinates of the new trapezoid are the same as the x-coordinates of the original trapezoid.

Question 6.
A triangle with coordinates (6, 4), (2, −1), and (−3, 5) is translated 4 units left and rotated 180° about the origin. What are the coordinates of its image?
Options:
a. (2, 4), (-2, -1), (-7, 5)
b. (4, 6), (-1, 2), (5, -3)
c. (4, -2), (-1, 2), (5, 7)
d. (-2, -4), (2, 1), (7, -5)

d. (-2, -4), (2, 1), (7, -5)

Question 7.
A rectangle with vertices (3, -2), (3, -4), (7, -2), (7, -4) is reflected across the x-axis and then rotated 90° counterclockwise.
a. In what quadrant does the image lie?
____________

After reflection and rotation, the image lies in QII

Question 7.
b. What are the vertices of the image?
Type below:
____________

image vertices
(-2, 3)
(-4, 3)
(-2, 7)
(-4, 7)

Question 7.
c. What other transformations produce the same image?
Type below:
____________