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