A rotation is a transformation where every point moves through the same angle about a fixed point − called the centre of rotation.

### Notation

**R** is a rotation of about **centre of rotation ****O**.

R: A A´ orR(A) = A´

- The
**angle of rotation**is the angle between the line joining the centre to a point, and the line joining the centre to the image of the point. - For anti-clockwise rotations, the angle of rotation is
**positive.** - For clockwise rotations the angle is
**negative.**

Properties of Rotation

Triangle PQR maps to triangle P´Q´R´ under a rotation of **°**, centre O.

- All points and lines, except the centre, turn through an angle of
**°**e.g. POP´ = QOQ´ = ROR´ = q

- Length, angle size and area are invariant.
e.g. length PQ = length P´Q´

- PQR = P´Q´R´
area PQR = area P´Q´R´

- PQR = P´Q´R´
- The centre of rotation is the only invariant point.
- Rotation is a direct transformation.
i.e. PQR and P´Q´R´ are both labelled in the anti-clockwise direction.

- Rotation is an isometry.
i.e. PQR is congruent to P´Q´R´.

### Location of Centre of Rotation

Given a figure and its image, to find the centre of the rotation:

1. Join any point and its image, and draw the **perpendicular bisector** of this line.

2. Repeat for another point and its image.

3. The centre of rotation is the intersection of the **mediators** (the perpendicular bisectors).

Download an interactive spreadsheet (Microsoft Excel) showing reflections, **rotations**, translations and enlargements.

(Windows users, right click and "Save target as..." to save the files on your computer.)