## Rotation

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´ or R(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´

• 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´.

See examples of rotations − ### 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). 