ROTATION
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Goals
Identify rotations in the plane. Apply rotation to figures on the
coordinate plane.
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Rotation
A transformation in which a figure is turned about a fixed point, called the center of rotation.
Center of Rotation
The center of rotation could be a point outside the shape or on the shape
A ROTATION MEANS TO TURN A FIGURE
A ROTATION MEANS TO TURN A FIGURE
center of rotation
ROTATION
A ROTATION MEANS TO TURN A FIGURE
A ROTATION MEANS TO TURN A FIGURE
The triangle was rotated around the point.
center of rotation
ROTATIONIf a shape spins
360, how far does it spin?
360
ROTATIONIf a shape spins
180, how far does it spin?
180
Rotating a shape 180 turns a shape upside
down.
ROTATIONIf a shape spins 90,
how far does it spin?
90
ROTATIONDescribe how the triangle A was transformed to
make triangle B
A B
Describe the translation.Triangle A was rotated right 90Triangle A was rotated right 90
ROTATIONDescribe how the arrow A was transformed to make
arrow B
Describe the translation.Arrow A was rotated right 180 Arrow A was rotated right 180
A
B
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Rotation
Rays drawn from the center of rotation to a point and its image form an angle called the angle of rotation.
Center of Rotation
90
G
G’
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A Rotation is an Isometry (Rigid Transformation)
Segment lengths are preserved.
Angle measures are preserved.
Parallel lines remain parallel. Orientation is unchanged.
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Rotations on the Coordinate Plane
Be able to do:
•90 rotations
•180 rotations
•clockwise & counter-clockwise
Unless told otherwise, the center of rotation is the origin (0, 0).
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90 clockwise rotation
Formula
(x, y) (y, x)A(-2, 4)
A’(4, 2)
Or…
Use the relation between the slopes of two perpendicular lines
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Rotate (-3, -2) 90 clockwise
Formula
(x, y) (y, x)
(-3, -2)
A’(-2, 3)
Or…
Use the relation between the slopes of two perpendicular lines
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90 counter-clockwise rotation
Formula
(x, y) (y, x)
A(4, -2)
A’(2, 4)
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Rotate (-5, 3) 90 counter-clockwise
Formula
(x, y) (y, x)
(-3, -5)
(-5, 3)
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180 rotation
Formula
(x, y) (x, y)
A(-4, -2)
A’(4, 2)
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Rotate (3, -4) 180
Formula
(x, y) (x, y)
(3, -4)
(-3, 4)
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Rotation Example
Draw a coordinate grid and graph:
A(-3, 0)
B(-2, 4)
C(1, -1)
Draw ABC
A(-3, 0)
B(-2, 4)
C(1, -1)
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Rotation Example
Rotate ABC 90 clockwise.
Formula
(x, y) (y, x)A(-3, 0)
B(-2, 4)
C(1, -1)
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Rotate ABC 90 clockwise.
(x, y) (y, x)
A(-3, 0) A’(0, 3)
B(-2, 4) B’(4, 2)
C(1, -1) C’(-1, -1)A(-3, 0)
B(-2, 4)
C(1, -1)
A’B’
C’
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Rotate ABC 90 clockwise.
Check by rotating ABC 90.
A(-3, 0)
B(-2, 4)
C(1, -1)
A’B’
C’
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Rotation Formulas 90 CW (x, y) (y, x) 90 CCW (x, y) (y, x) 180 (x, y) (x, y) These rules only work when the
center of rotation is the origin. Use the opposite reciprocal relation between the slopes of perpendicular lines to do rotations about other points.
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Rotating segments
A
B
C
D
E
F
G
H
O
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Rotating AC 90 CW about the origin maps it to _______.
A
B
C
D
E
F
G
H
CE
O
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Rotating HG 90 CCW about the origin maps it to _______.
A
B
C
D
E
F
G
H
FE
O
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Rotating AH 180 about the origin maps it to _______.
A
B
C
D
E
F
G
H
ED
O
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Rotating GF 90 CCW about point G maps it to _______.
A
B
C
D
E
F
G
H
GH
O
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Rotating ACEG 180 about the origin maps it to _______.
A
B
C
D
E
F
G
H
EGAC
A E
C
G
O
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Rotating FED 270 CCW about point D maps it to _______.
A
B
C
D
E
F
G
H
BOD
O
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Summary
A rotation is a transformation where the preimage is rotated about the center of rotation.
Rotations are Rigid Transformations
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Homework