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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Geometry

9-3 Rotations

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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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90 clockwise rotation

Formula

(x, y) (y, x)

A(-2, 4)

A’(4, 2)

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Rotate (-3, -2) 90 clockwise

Formula

(x, y) (y, x)

(-3, -2)

A’(-2, 3)

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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)

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

Homework

Page 644 #’s 14-18 Evens Only