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Page 1: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-1 Reteach to Build UnderstandingReflections

1. Tell whether each transformation is a rigid motion or is not a rigid motion.

This is , because the size and shape do not change.

This is , because the size changes.

2. Example: The graph shows the reflection of quadrilateral ABCD across line m. The reflection is written R m (ABCD) → (A′B′C′D′).

y

6

2

4

Ox

62

B

CD

A

B′

A′

C′ D′

m

Esteban said R n (ABC) → (A′B′C′), where the equation of line n is x = 4, is the rule for the reflection. What was his error?

y

6

2

4

Ox

4 62

CA

B

B′

A′ C′

Esteban used the wrong line of reflection. The image is reflected across the line y = 4.

3. Which is the line of reflection for each pair of figures?

a

b

a

b

a rigid motion

Line aLine b

not a rigid motion

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Page 2: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-1 Additional PracticeReflections

Tell whether the transformation appears to be a rigid motion. Explain.

1.

Preimage Image

Yes; angle measure and length are preserved.

2.

Preimage Image

Show the reflection of △ABC across line ℓ.

3.

A′

B′C′A

B

C

4.

A′

B′

C′

AB

C

5. Suppose the equation of line ℓ is x = 1. Given points M(3, 3), N(4, 4), and O(5, 2), graph △MNO and the reflection image R ℓ (△MNO).

6. Understand What is the reflection rule for the triangle and image with coordinates A(2, 4), B(4, 6), C(5, 2), and A′(−4, −2), B′(−6, −4), C′(−2, −4)? R m (x, y) = (−y, −x), where the equation of line m is y = −x

7. Apply Student A sits in a chair facing a mirror and sees the reflection image B′ of Student B in the mirror. Show the actual position of Student B.

y

2

Ox

4 62−4 −2

−2

O

NN′M′

O′

M

B A

B′

mirror

No; angle measure is preserved, but the image is smaller than the preimage, so there is a change in length.

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Page 3: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-2 Reteach to Build UnderstandingTranslations

1. Example: The displacement in the translation is (4, –5), so the image is moved 4 units in the positive x-direction and 5 units in the negative y-direction.

T ⟨4, −5⟩ (△MNP)

y4

2

Ox

42−4 −2

−2

M

NP

M′

N′P′

Which is the image of JKL?

T ⟨−2, 1⟩ (△JKL)

y4

2

x−4 −2

−2

B

AC

K

JL

E

DF

2. Deshawn wrote the following rule for the translation shown. What was his error? T ⟨−4, −3⟩ (△JKL)

y

Ox

42−4

−4

K

JL

K′

J′L′

The image moved three units left and four units down. Deshawn should have written T ⟨−3, −4⟩ (△JKL), because the translation is ⟨ −3, −4 ⟩ .

3. Give the coordinates of the image. T ⟨3, −2⟩ (△ABC) for A(4, 1), B(−3, 2), C(4, −5)

A′(7, −1), B’ , C’

T ⟨−5, 0⟩ (△DEF) for D(4, 4), E(−3, 5), F(0, 7)

D′(−1, 4), E′ , F′

T ⟨−8, −5⟩ (△GHJ) for G(0, 0), H(4, 3), J(9, 7)

G′(−8, −5), H′ , J′

△ABC

(−4, −2)

(7, −7)

(−8, 5)

(0, 0)

(1, 2)

(−5, 7)

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Page 4: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-2 Additional PracticeTranslations

What is the rule for the translation shown?

1. y

Ox

−4 −2

−2

A B

CD

A′ B′

C′D′

T ⟨5, −2⟩

2. y

Ox

−4 −2 2 4

4

R

Q

PS

R′

Q′

P′S′

T ⟨2, −1⟩

The vertices of △ABC are A(2, −3), B(−3, −5), and C(4, 1). For each translation, give the vertices of △A′B′C′.

3. T ⟨−2, 3⟩ (△ABC) 4. T ⟨−4, −1⟩ (△ABC) 5. T ⟨4, 6⟩ (△ABC)

Write the composition of transformations as one transformation.

6. T ⟨4, 5⟩ ∘ T ⟨3, 1⟩ 7. T ⟨−1, −3⟩ ∘ T ⟨2, −2⟩ 8. T ⟨1, 1⟩ ∘ T ⟨−4, −3⟩

Given △XYZ with vertices X(−2, 1), Y(−1, 3), and Z(−4, 2), write the translation equivalent to the composition of transformations. Suppose the equation of line m is x = 5, the equation of line n is y = 4, and the equation of line p is x = 3.

9. R m ∘ R y-axis 10. R n ∘ R x-axis 11. R p ∘ R y-axis

12. Understand How far apart are two parallel lines ℓ and m such that T ⟨4, 0⟩ (△DEF) = ( R m ∘ R ℓ )(△DEF)?

13. Apply The composition of rigid motions T ⟨10, 2⟩ ∘ T ⟨−23, −3⟩ describes the route of a limousine in New York City from its starting position. How would you describe the route in words?

A′(0, 0), B′(−5, −2), C′(2, 4)

Sample answer: The limousine drives 23 blocks west and 3 blocks south and then 10 blocks east and 2 blocks north.

2 units

T ⟨7, 6⟩

T ⟨10, 0⟩ (△XYZ) T ⟨0, −8⟩ (△XYZ)

A′(−2, −4), B′(−7, −6), C′(0, 0)

T ⟨1, −5⟩

A′(6, 3), B′(1, 1), C′(8, 7)

T ⟨−3, −2⟩

T ⟨6, 0⟩ (△XYZ)

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Page 5: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-3 Reteach to Build UnderstandingRotations

1. A line drawn from N to P to N′ makes a 90° angle.

Example:

r (90°, P)( ‾ MN )

N′

M′

M

N

P

Which is the rotated image?

r (180°, T)(△ABC)

B

A

C

T

H

G

I

K

JL

E

DF

2. If A(3, 7) and B(−3, 4), what is r (90°, O)( ‾ AB )? Chris found the coordinates of ‾ A′B′ . What was his error?

A′(7, −3), B′(4, 3) Chris rotated the image 90° clockwise rather than counterclockwise.

3. The diagram shows r (245°, P) (△ABC) → △A″B″C″. How can the rotation be accomplished as the composition of two reflections?

B

A

C

m

n

P

B″B′

A″

A′

C′ = C″

The rotation can be done by constructing a line n through the center of rotation and then reflecting △ABC across line n. Next, construct line m containing the midpoint of ‾ B′B″ and the , and reflect the intermediate image △A′B′C′ across that line to obtain the final image △A″B″C″ .

center of rotation

△GHI

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Page 6: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-3 Additional PracticeRotations

1. Draw the rotated image. r (270°, A) (△XYZ)

For Exercises 2 and 3, give the coordinates of each image.

2. r (90°, O) (△MN) for M(3, −5), N(2, 4)

3. r (180°, O) (△ABC) for A(1, 1), B(3, 5), C(5, 2)

4. Understand Draw two lines of reflection so that the composition of the reflections across the lines maps onto the image shown.

5. Apply A blender has blades as shown. What rotation will map the blade formed by △ABC onto the blade formed by △DEF?

A′(−1, −1), B′(−3, −5), C′(−5, −2)

180°

M′(5, 3), N′(−4, 2)

AXY

ZX′ Z′

Y′

A

K

LJL′

a b

K′

J′

D

E

C

A

B

F

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Page 7: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-4 Reteach to Build UnderstandingClassification of Rigid Motions

1. A glide reflection is a composition of a reflection followed by a translation. Given A(5, −4), B(2, 5), C(−3, 4), draw △A′B′C′ for each glide reflection.

Example: (T ⟨0, −2⟩ ∘ R p)(△ABC) = △A′B′C′

y

Ox

42−4 −2−8 −6

−6

−4

−2

4

6

86

2

B

B′C′

C

A′

A

p

(T ⟨4, 0⟩ ∘ R q)(△ABC) = △A′B′C′

Which shows △A′B′C′?

y

Ox

−4

−2

6

8

2

8

B

A

HI

C

FE

q

D

G LK

J

2. Carolina wrote the following rule for the glide reflection that maps △MNO to △M′N′O′: (Rℓ ∘ T⟨0, −4⟩)(△MNO) = △M′N′O′.

Explain why should she have written (T⟨0, −4⟩ ∘ Rℓ)(△MNO) = △M′N′O′.

3. A glide reflection is a composition of that first the preimage across a line and then the

image in the direction of the line. A glide reflection is written T⟨x, y⟩ ∘ Rm, with the operation to perform on the right and the operation to perform on the left.

△JKL

y

Ox

42−4 −2−8 −6

−6

−4

−2

4

86

2

8

−8

M

Q′

M′

N′

N

Q

Carolina wrote the composition of the transformations backwards. The translation should be written first.

rigid motionsreflects translates

firstsecond

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Page 8: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-4 Additional PracticeClassification of Rigid Motions

For Exercises 1–5, use the diagram.

1. What composition of two rigid motions maps △ABC to △A′B′C′?

For Exercises 2–5, find the coordinates of P ′ under each transformation. Suppose the equation of line m is y = 2 and the equation of line n is x = − 1.

2. T ⟨−2, 0⟩ ∘ R m 3. T ⟨0, −5⟩ ∘ R n

4. T ⟨0, 2⟩ ∘ R y-axis 5. T ⟨3, 0⟩ ∘ R x-axis

For Exercises 6–12, describe the rigid motion that produces each image.

6. △ABC → △DEF

7. △ABC → △GHJ

8. △ABC → △KLM

9. △ABC → △NPQ

10. △ABC → △RST

11. △DEF → △GHJ

12. △GHJ → △KLM

13. Understand Define the term glide reflection. A glide reflection is a reflection across a line followed by a translation in the direction of the line.

14. Apply The series of footprints can be described as a series of glide reflections. The compositon of two identical glide reflections, for example, from the first step to the third, is equivalent to what rigid motion? a translation with twice the displacement of the glide reflection

y

2

Ox

42−4 −2−2

−4 BC

A

PA′

C′ B′

y

2

4

O

x

4

−2

−4B

E

P

N

Q

G K

M L

ST

R

H J

C

A

D

F

T ⟨−5, 0⟩ ∘ R x-axis

T ⟨−4, 0⟩ ∘ R y-axis

T ⟨−1, 0⟩ ∘ R x-axis

T ⟨−6, 0⟩ ∘ R y = −2

T ⟨−6, 4⟩

R y = −1

R x = 0.5

(−1, 3)

r (180°, (−3, 2))

(3, −1) (0, −1)

(−3, −4)

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Page 9: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-5 Reteach to Build UnderstandingSymmetry

1. A line of symmetry divides a figure into two congruent halves. For each figure, draw the lines of symmetry.

a. The figure has lines of symmetry.

b. The figure has lines of symmetry.

c. The figure has lines of symmetry.

d. The figure has lines of symmetry.

2. Alberto listed 90°, 180° and 270° as the angles of rotation for which the figure has rotational symmetry. What was his error?Alberto did not list the lines of symmetry that connect opposite vertices of the octagon. He should have listed 45°, 90°, 135°, 180°, 225°, 270°, and 315°.

3. List the angles of rotation for which the figures have rotational symmetry.

a. 90°, , and b. and 240°

6

2

4

5

180° 270° 120°

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Page 10: Name...Name PearsonRealize.com 3-1 Additional Practice Reflections Tell whether the transformation appears to be a rigid motion. Explain. 2.1. Preimage Image Yes; angle measure and

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3-5 Additional PracticeSymmetry

For Exercises 1–6, draw all lines of symmetry for each figure, or write “None” in the figure.

1. 2. 3.

4. 5.

None

6.

For Exercises 7–9, find every angle of rotation that maps the figure onto itself, or write “None” in the figure.

7.

45°, 90°, 135°, 180°,225°, 270°, 315°

8.

90°, 180°, 270°

9.

180°

10. Understand What does it mean for a figure to have reflectional symmetry? Rotational symmetry?

For a figure with reflectional symmetry, there are one or more lines over which the figure can be reflected to map onto itself. For rotational symmetry, there are one or more angles of rotation about the center that will map the figure onto itself.

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