Geometry

Rotations

04/21/23

Goals

Identify rotations in the plane. Apply rotation formulas to figures

on the coordinate plane.

04/21/23

Rotation

A transformation in which a figure is turned about a fixed point, called the center of rotation.

Center of Rotation

04/21/23

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’

04/21/23

A Rotation is an Isometry

Segment lengths are preserved. Angle measures are preserved. Parallel lines remain parallel. Orientation is unchanged.

04/21/23

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).

04/21/23

90 clockwise rotation

Formula

(x, y) (y, x)A(-2, 4)

A’(4, 2)

04/21/23

Rotate (-3, -2) 90 clockwise

Formula

(x, y) (y, x)

(-3, -2)

A’(-2, 3)

04/21/23

90 counter-clockwise rotation

Formula

(x, y) (y, x)

A(4, -2)

A’(2, 4)

04/21/23

Rotate (-5, 3) 90 counter-clockwise

Formula

(x, y) (y, x)

(-3, -5)

(-5, 3)

04/21/23

180 rotation

Formula

(x, y) (x, y)

A(-4, -2)

A’(4, 2)

04/21/23

Rotate (3, -4) 180

Formula

(x, y) (x, y)

(3, -4)

(-3, 4)

04/21/23

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)

04/21/23

Rotation Example

Rotate ABC 90 clockwise.

Formula

(x, y) (y, x)A(-3, 0)

B(-2, 4)

C(1, -1)

04/21/23

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’

04/21/23

Rotate ABC 90 clockwise.

Check by rotating ABC 90.

A(-3, 0)

B(-2, 4)

C(1, -1)

A’B’

C’

04/21/23

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.

04/21/23

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.

04/21/23

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

04/21/23

Compound Reflections Furthermore, the amount of the rotation is

twice the measure of the angle between lines k and m.

P

mk

45

90

04/21/23

Compound Reflections The amount of the rotation is twice the

measure of the angle between lines k and m.

P

mk

x

2x

04/21/23

Rotational Symmetry

A figure can be mapped onto itself by a rotation of 180 or less.

4590

The square has rotational symmetry of 90.

04/21/23

Does this figure have rotational symmetry?

The hexagon has rotational symmetry of 60.

04/21/23

Does this figure have rotational symmetry?

Yes, of 180.

04/21/23

Does this figure have rotational symmetry?

No, it required a full 360 to map onto itself.

90

180

270360

04/21/23

Rotating segments

A

B

C

D

E

F

G

H

O

04/21/23

Rotating AC 90 CW about the origin maps it to _______.

A

B

C

D

E

F

G

H

CE

O

04/21/23

Rotating HG 90 CCW about the origin maps it to _______.

A

B

C

D

E

F

G

H

FE

O

04/21/23

Rotating AH 180 about the origin maps it to _______.

A

B

C

D

E

F

G

H

ED

O

04/21/23

Rotating GF 90 CCW about point G maps it to _______.

A

B

C

D

E

F

G

H

GH

O

04/21/23

Rotating ACEG 180 about the origin maps it to _______.

A

B

C

D

E

F

G

H

EGAC

A E

C

G

O

04/21/23

Rotating FED 270 CCW about point D maps it to _______.

A

B

C

D

E

F

G

H

BOD

O

04/21/23

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.

04/21/23

Homework