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Rotations

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Rotations

- Identify rotations and rotational symmetry.

- Rotation
- Center of rotation
- Angle of rotation
- Rotational symmetry

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

image

Pre-image

rotation

fixed point

- Center of rotation – fixed point of the rotation.

Center of Rotation

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

image

Pre-image

Angle of Rotation

Example:

Click the

triangle to

see rotation

Center of Rotation

Rotation

Example 1: Identifying Rotations

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

B.

A.

No; the figure appears

to be flipped.

Yes; the figure appears

to be turned around a point.

Your Turn:

Tell whether each transformation appears to be a rotation.

b.

a.

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

No, the figure appears to be a translation.

- 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

3

An equilateral triangle has rotational symmetry of order ?

3

2

1

Regular Hexagon

A regular hexagon has rotational symmetry of order ?

Regular Hexagon

A regular hexagon has rotational symmetry of order ?

Regular Hexagon

6

A regular hexagon has rotational symmetry of order ?

5

6

1

4

2

3

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

Symmetry

Rotational:120°90°60°45°

a.

Rectangle

b.

Regular hexagon

c.

Trapezoid

SOLUTION

a.

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

Example 2

Identify Rotational Symmetry

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

b.

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

c.

No. A trapezoid does not have rotational symmetry.

Example 2

Identify Rotational Symmetry

Regular hexagon

Trapezoid

Your Turn:

1.

Isosceles trapezoid

no

ANSWER

2.

Parallelogram

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

ANSWER

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

Your Turn:

3.

Regular octagon

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

ANSWER

- 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

8

7

6

y

5

4

3

2

1

x

x

x

x

1 2 3 4 5 6 7 8

–7 –6 –5 –4 –3 –2 –1

-1

x

-2

-3

-4

-5

-6

x

A Rotation of 90° Counterclockwiseabout (0,0)

(x, y)→(-y, x)

C(3,5)

B’(-2,4)

C’(-5,3)

B(4,2)

A’(-1,2)

A(2,1)

8

7

6

y

5

4

3

2

1

x

x

x

x

x

x

x

1 2 3 4 5 6 7 8

–7 –6 –5 –4 –3 –2 –1

-1

x

-2

-3

-4

-5

-6

x

A Rotation of 180° about (0,0)

(x, y)→(-x, -y)

C(3,5)

B(4,2)

A(2,1)

A’(-2,-1)

B’(-4,-2)

C’(-3,-5)

SOLUTION

Plot the points, as shown in blue.

Example 4

Rotations in a Coordinate Plane

Sketch the quadrilateral with vertices A(2, –2),B(4, 1), C(5, 1), andD(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 the

rotated vertices.

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

Checkpoint

4.

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

ANSWER

Rotations in a Coordinate Plane

- A'(0,0), B'(0, 3), C'(–4, 3)