Transformations 8th Grade Math 2D Geometry: Transformations

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New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

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Table of Contents

? Transformations ? Translations ? Rotations ? Reflections ? Dilations ? Symmetry ? Congruence & Similarity ? Special Pairs of Angles

Click on a topic to go to that section

Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5

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Any time you move, shrink, or enlarge a figure you make a transformation. If the figure you are moving (pre-image) is labeled with letters A, B, and C, you can label the points on the transformed image (image) with the same letters and the prime sign.

B A

B' A'

pre-image

image

C C'

for transformation shown

Pull

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8th Grade Math 2D Geometry: Transformations

2013-12-09

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Transformations

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The image can also be labeled with new letters as shown below. Triangle ABC is the pre-image to the reflected image triangle XYZ

B A

pre-image

Y X

image

CZ

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There are four types of transformations in this unit:

? Translations ? Rotations ? Reflections ? Dilations

The first three transformations preserve the size and shape of the figure. They will be congruent. Congruent figures are same size and same shape.

In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid.

If your pre-image is an angle, your image is an angle with the same measure.

If your pre-image contains parallel lines, your image contains parallel lines.

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There are four types of transformations in this unit:

? Translations ? Rotations ? Reflections ? Dilations

The first three transformations preserve the size and shape of the figure.

In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid.

If your pre-image is an angle, your image is an angle with the same measure.

If your pre-image contains parallel lines, your image contains parallel lines.

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Translations

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A translation is a slide that moves a figure to a different position (left, right, up or down) without changing its size or shape and without flipping or turning it.

You can use a slide arrow to show the direction and distance of the movement.

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This shows a translation of pre-image ABC to image A'B'C'. Each point in the pre-image was moved right 7 and up 4.

PULL

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Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image?

A C

B

PULL

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To complete a translation, move each point of the pre-image and label the new point.

Example: Move the figure left 2 units and up 5 units. What are the coordinates of the pre-image and image?

A'

B'

D' A

D

C' B

C

Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

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Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image?

A B

D

C

PULL

Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

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Translate pre-image ABCD 5 left and 3 up. What are the coordinates of the image and pre-image?

A B

C D

Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

PULL

Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved? Both the pre-image and image are congruent.

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A rule can be written to describe translations on the coordinate plane. Look at the following rules and coordinates to see if you can find a pattern.

2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1)

2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3)

5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1)

4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

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Translating left/right changes the x-coordinate. Translating up/down changes the y-coordinate.

2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1)

5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1)

2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3)

4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

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Translating left/right changes the x-coordinate. ? Left subtracts from the x-coordinate ? Right adds to the x-coordinate

Translating up/down changes the y-coordinate. ? Down subtracts from the y-coordinate ? Up adds to the y-coordinate

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A rule can be written to describe translations on the coordinate plane.

2 units Left ...

x-coordinate - 2 y-cocolircdkintoatreevsetaayl s

rule = (x - 2, y)

5 units Right & 3 units Down...

x-coordinate + 5 y-ccoloicrdkintoatreev-e3al rule = (x + 5, y - 3)

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Write a rule for each translation.

2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1)

(xc,liyc)k to(xre-2v,eya+l 5)

2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3)

(xc,lyic)k to(xr-e2v,eya-6l )

5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1)

(x,cyli)ck t(ox-r5e,vye+a3l )

4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

(xc, lyic) k to(xr+e4v,eya-l1)

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1 What rule describes the translation shown?

2 What rule describes the translation shown?

A (x,y) (x - 4, y - 6) B (x,y) (x - 6, y - 4)

D' E

E' F'

A (x,y) (x, y - 9)

B (x,y) (x, y - 3)

E

C (x,y) (x + 6, y + 4) D (x,y) (x + 4, y + 6)

D

F

G'

C (x,y) (x - 9, y) Pull

D (x,y) (x - 3, y)

D

F

Pull

G

G

E' D'

F'

G'

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3 What rule describes the translation shown?

4 What rule describes the translation shown?

A (x,y) (x + 8, y - 5)

A (x,y) (x - 3, y + 2)

B (x,y) (x - 5, y - 1)

E'

Pull

B (x,y) (x + 3, y - 2)

E

Pull

C (x,y) (x + 5, y - 8)

D'

D (x,y) (x - 8, y + 5)

F' E

C (x,y) (x + 2, y - 3) D (x,y) (x - 2, y + 3)

D D' F E' F'

D

G'

F

G

G'

G

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5 What rule describes the translation shown?

A (x,y) B (x,y) C (x,y) D (x,y)

(x - 3, y + 2) (x + 3, y - 2) (x + 2, y - 3) (x - 2, y + 3)

E'

Pull

D'

F'

E

D

F

G'

G

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Rotations

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A rotation (turn) moves a figure around a point. This point can be on the figure or it can be some other point. This point is called the point of rotation.

P

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