how to find the centre of rotation

Unit 7 Section 5 : Rotations

In this section we review rotational symmetry and draw rotations of shapes.

Example 1

State the order of rotational symmetry of each of the following shapes:

(a)

Order 4. This means that the shape can be rotated 4 times about its centre before returning to its starting position. Each rotation will be through an angle of 90°, and, after each one, the rotated shape will occupy the same position as the original square.

(b)

Order 2

(c)

Order 1. This means that the shape does not have rotational symmetry.

Example 2

The corners of a rectangle have coordinates (3, 2), (7, 2), (7, 5) and (3, 5). The rectangle is to be rotated through 90 ° clockwise about the origin.
Draw the original rectangle and its position after being rotated.

The following diagram shows the original rectangle A B C D and the rotated rectangle A' B' C' D'. The curves show how each corner moves as it is rotated. The easiest way to rotate a shape is to place a piece of tracing paper over the shape, trace the shape, and then rotate the tracing paper about the centre of rotation, as shown.

Example 3

A triangle has corners at the points with coordinates (4, 7), (2, 7) and (4, 2).

(a)

Draw the triangle.

(b)

Rotate the triangle through 180° about the point (4, 1).

The diagram shows how the original triangle A B C is rotated about the point (4, 1) to give the triangle A' B' C'.

Example 4

The diagram shows the triangle A B C which is rotated through 90° to give A' B' C'.
Determine the position of the centre of rotation.

The first step is to join the points A and A' and draw the perpendicular bisector of this line.
The centre of rotation must be on this line.

Repeat the process, drawing the perpendicular bisectors of B B' and C C' as shown opposite.
The point where the lines cross is the centre of rotation.

Note:

For simple rotations you may be able to spot the centre of rotation without having to use the method shown above. Alternatively, you may be able to find the centre of rotation by experimenting with tracing paper.

Exercises

Question 12

(a)

You can rotate triangle A onto triangle B. Make a copy of the diagram and put a cross on the centre of rotation. You may use tracing paper to help you.

(b)

You can rotate triangle A onto triangle B. The rotation is anti-clockwise. What is the angle of rotation?

(c)

On the diagram below, reflect triangle A in the mirror line. You may use a mirror or tracing paper to help you.

Question 13

Julie has written a computer program to transform pictures of tiles. There are only two instructions in her program,

reflect vertical

rotate 90° clockwise.

(a)

Julie wants to transform the first pattern to the second pattern.

Complete the following instructions to transform the tiles B1 and B2. You must use only reflect vertical or rotate 90° clockwise.

A1	Tile is in the correct position.
A2	Reflect vertical, and then rotate 90° clockwise.
B1	Rotate 90° clockwise and then
B2

(b)

Paul starts with the first pattern that was on the screen.

Complete the instructions for the transformations of A2, B1 and B2 to make Paul's pattern. You must use only reflect vertical or rotate 90° clockwise.