Geometry - BakerMath.org
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Transcript Geometry - BakerMath.org
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’
A(-3, 0)
B’
(x, y) (y, x)
A(-3, 0) A’(0, 3)
B(-2, 4) B’(4, 2)
C’ C(1, -1)
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C(1, -1) C’(-1, -1)
Rotate ABC 90 clockwise.
B(-2, 4)
A’
A(-3, 0)
B’
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. 4/29/2020
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
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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