Review from Friday
The composition of two reflections over parallel lines can be described by a translation vector that is: Perpendicular to the two lines Twice the distance between the two
lines
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04/21/23
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.
Center of Rotation
90
G
G’
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A Rotation is an Isometry
Segment lengths are preserved
Angle measures are preserved Parallel lines remain parallel
Rotations on the Coordinate Plane
Know the formulas for:
•90 rotations
•180 rotations
•clockwise & counter-clockwise
Unless told otherwise, the center of rotation is the origin (0, 0).
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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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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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Rotation Formulas
90 CW (x, y) (y, x) 90 CCW (x, y) (y, x) 180 (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.
P
mk
Compound Reflections Furthermore, the amount of the
rotation is twice the measure of the angle between lines k and m.
P
mk
45
90
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Compound Reflections The amount of the rotation is twice
the measure of the angle between lines k and m.
P
mk
x
2x
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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