Geometry
Rotations
Goals


Identify rotations in the plane.
Apply rotation formulas 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
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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.
G
90
Center of Rotation
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G’
A Rotation is an Isometry




Segment lengths are preserved.
Angle measures are preserved.
Parallel lines remain parallel.
Orientation is unchanged.
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Rotations on the Coordinate Plane
Know the formulas for:
•90 rotations
•180 rotations
•clockwise & counterclockwise
Unless told otherwise, the center of rotation is the origin (0, 0).
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90 clockwise rotation
A(-2, 4)
Formula
(x, y)  (y, x)
A’(4, 2)
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Rotate (-3, -2) 90 clockwise
Formula
A’(-2, 3)
(-3, -2)
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(x, y)  (y, x)
90 counter-clockwise rotation
Formula
A’(2, 4)
(x, y)  (y, x)
A(4, -2)
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Rotate (-5, 3) 90 counter-clockwise
Formula
(-5, 3)
(-3, -5)
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(x, y)  (y, x)
180 rotation
Formula
(x, y)  (x, y)
A’(4, 2)
A(-4, -2)
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Rotate (3, -4) 180
Formula
(-3, 4)
(x, y)  (x, y)
(3, -4)
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Rotation Example
B(-2, 4)
Draw a coordinate
grid and graph:
A(-3, 0)
A(-3, 0)
C(1, -1)
B(-2, 4)
C(1, -1)
Draw ABC
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Rotation Example
B(-2, 4)
Rotate ABC 90
clockwise.
Formula
A(-3, 0)
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C(1, -1)
(x, y)  (y, x)
Rotate ABC 90 clockwise.
B(-2, 4)
A’
B’
A(-3, 0) C’
C(1, -1)
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(x, y)  (y, x)
A(-3, 0)  A’(0, 3)
B(-2, 4)  B’(4, 2)
C(1, -1)  C’(-1, -1)
Rotate ABC 90 clockwise.
B(-2, 4)
A’
B’
A(-3, 0) C’
C(1, -1)
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Check by rotating
ABC 90.
Rotation Formulas




90 CW
90 CCW
180
(x, y)  (y, x)
(x, y)  (y, x)
(x, y)  (x, y)
Rotating through an angle other than
90 or 180 requires much more
complicated math.
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Compound Reflections

If lines k and m intersect at point P,
then a reflection in k followed by a
reflection in m is the same as a rotation
about point P.
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Compound Reflections

If lines k and m intersect at point P, then a reflection in k
followed by a reflection in m is the same as a rotation about
point P.
k
m
P
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Compound Reflections

Furthermore, the amount of the rotation is twice the
measure of the angle between lines k and m.
k
m
45
90
P
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Compound Reflections

The amount of the rotation is twice the measure of
the angle between lines k and m.
k
m
x
2x
P
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Rotational Symmetry

A figure can be mapped onto itself by a
rotation of 180 or less.
45
90
The square has rotational symmetry of
90. 4/13/2015
Does this figure have
rotational symmetry?
The hexagon has rotational symmetry of 60.
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Does this figure have
rotational symmetry?
Yes, of 180.
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Does this figure have
rotational symmetry?
90
180
270
360
No, it required a full 360 to
map onto itself.
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Rotating segments
C
B
A
D
O
H
F
G
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E
Rotating AC 90 CW about the
origin maps it to _______.
CE
C
B
A
D
O
H
F
G
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E
Rotating HG 90 CCW about
the origin maps it to _______.
FE
C
B
A
D
O
H
F
G
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E
Rotating AH 180 about the
origin maps it to _______.
ED
C
B
A
D
O
H
F
G
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E
Rotating GF 90 CCW about
point G maps it to _______.
GH
C
B
A
D
O
H
F
G
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E
Rotating ACEG 180 about the
origin maps it to _______.
EGAC
C
C
B
A A
D
O
H
F
G
G
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E E
Rotating FED 270 CCW about
BOD
point D maps it to _______.
C
B
A
D
O
H
F
G
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E
Summary
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

A rotation is a transformation where the
preimage is rotated about the center of
rotation.
Rotations are Isometries.
A figure has rotational symmetry if it
maps onto itself at an angle of rotation
of 180 or less.
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Homework
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