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3-1 Reteach to Build Understanding Reflections

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

O x

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

O x

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 a Line b

not a rigid motion

enVision™ Geometry • Teaching Resources

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3-1 Additional Practice Reflections

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

ℓ

4.

A′

B′

C′

A B

C

ℓ

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

O x

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.

enVision™ Geometry • Teaching Resources

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3-2 Reteach to Build Understanding Translations

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)

y 4

2

O x

42−4 −2 −2

M

N P

M′

N′ P′

Which is the image of JKL?

T 〈−2, 1〉 (△JKL)

y 4

2

x −4 −2

−2

B

A C

K

J L

E

D F

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

y

O x

42−4

−4

K

J L

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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3-2 Additional Practice Translations

What is the rule for the translation shown?

1. y

O x

−4 −2 −2

A B

C D

A′ B′

C′ D′

T 〈5, −2〉

2. y

O x

−4 −2 2 4

4

R

Q

P S

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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3-3 Reteach to Build Understanding Rotations

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

J L

E

D F

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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3-3 Additional Practice Rotations

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)

A XY

Z X′ Z′

Y′

A

K

LJ L′

a b

K′

J′

D

E

C

A

B

F

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3-4 Reteach to Build Understanding Classification 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

O x

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

O x

−4

−2

6

8

2

8

B

A

H I

C

F E

q

D

G L K

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