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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Click for web page
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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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