Rotations

Goal

• Identify rotations and rotational symmetry.

Key Vocabulary

• Rotation• Center of rotation• Angle of rotation• Rotational symmetry

Rotation Vocabulary

• Rotation – transformation that turns every point of a pre-image through a specified angle and direction about a fixed point.

Pre-imageimage

fixed point

rotation

Rotation Vocabulary

• Center of rotation – fixed point of the rotation.

Center of Rotation

Rotation Vocabulary

• Angle of rotation – angle between a pre-image point and corresponding image point.

Angle of Rotation

imagePre-image

Click the

triangle to

see rotationCenter of Rotation

Example:

Example 1: Identifying Rotations

Tell whether each transformation appears to be a rotation. Explain.

No; the figure appearsto be flipped.

Yes; the figure appearsto be turned around a point.

A. B.

Your Turn:

Tell whether each transformation appears to be a rotation.

No, the figure appears to be a translation.

Yes, the figure appears to be turned around a point.

a. b.

Rotation Vocabulary

• Rotational symmetry – A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180⁰ or less.

Has rotational symmetry because it maps onto itself by a rotation of 90⁰.

Equilateral Triangle

An equilateral triangle has rotational symmetry of order ?

Equilateral Triangle

An equilateral triangle has rotational symmetry of order ?

Equilateral Triangle

An equilateral triangle has rotational symmetry of order ?

12

33

Hexagon

Regular HexagonA regular hexagon has rotational symmetry of order ?

Regular HexagonA regular hexagon has rotational symmetry of order ?

1

23

4

5 6

Regular HexagonA regular hexagon has rotational symmetry of order ? 6

Rotational Symmetry

When a figure can be rotated less than 360° and the image and pre-image are indistinguishable (regular polygons are a great example).

SymmetryRotational: 120° 90° 60° 45°

Identify Rotational SymmetryExample 2

Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself.

Regular hexagonb.Rectanglea. Trapezoidc.

SOLUTION

Yes. A rectangle can be mapped onto itself by a clockwise or counterclockwise rotation of 180° about its center.

a.

Identify Rotational SymmetryExample 2

Yes. A regular hexagon can be mapped onto itself by a clockwise or counterclockwise rotation of 60°, 120°, or 180° about its center.

b.

No. A trapezoid does not have rotational symmetry.c.

Regular hexagon

Trapezoid

Your Turn:

Does the figure have rotational symmetry? If so, describe the rotations that map the figure onto itself.

Isosceles trapezoid1.

Parallelogram2.

noANSWER

yes; a clockwise or counterclockwise rotation of 180° about its center

ANSWER

Your Turn:

Regular octagon3.

yes; a clockwise or counterclockwise rotation of 45°, 90°, 135°, or 180° about its center

ANSWER

Rotation in a Coordinate Plane

• For a Rotation, you need;• An angle or degree of turn

– Eg 90° or a Quarter Turn– E.g. 180 ° or a Half Turn

• A direction– Clockwise– Anticlockwise

• A Centre of Rotation– A point around which Object rotates

y

x 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

–7 –6 –5 –4 –3 –2 –1 -1-2-3-4-5-6

A Rotation of 90° Counterclockwise about (0,0)

xxxxxA(2,1)

B(4,2)

C(3,5)

A’(-1,2)

B’(-2,4)

C’(-5,3)

(x, y)→(-y, x)

y

x 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

–7 –6 –5 –4 –3 –2 –1 -1-2-3-4-5-6

A Rotation of 180° about (0,0)

xxxxx

xxxA(2,1)

B(4,2)

C(3,5)

(x, y)→(-x, -y)

A’(-2,-1)B’(-4,-2)

C’(-3,-5)

Rotation in a Coordinate Plane

Rotations in a Coordinate PlaneExample 4

Sketch the quadrilateral with vertices A(2, –2), B(4, 1), C(5, 1), and D(5, –1). Rotate it 90° counterclockwise about the origin and name the coordinates of the new vertices.

Use a protractor and a ruler to find therotated vertices.

The coordinates of the vertices of the image are A'(2, 2), B'(–1, 4), C'(–1, 5), and D'(1, 5).

SOLUTION

Plot the points, as shown in blue.

Checkpoint Rotations in a Coordinate Plane

Sketch the triangle with vertices A(0, 0), B(3, 0), and C(3, 4). Rotate ∆ABC 90° counterclockwise about the origin. Name the coordinates of the new vertices A', B', and C'.

4.

ANSWER

A'(0, 0), B'(0, 3), C'(–4, 3)