Rotation in a plane
Definition and construction of a rotation
What is a rotation?
A rotation is a transformation of the plane that ‘rotates’ a figure around a fixed point, called the centre, by a certain angle, in a given direction (clockwise or anti-clockwise, also known as the trigonometric direction).
A rotation is defined by three parameters:
- a centre O
- an angle of rotation (in degrees)
- a direction (the trigonometric direction, which is positive, is anti-clockwise)
The rotation with centre O and angle theta is often denoted by R(O, theta).
How do we construct the image of a point?
To construct the image M’ of a point M by rotation about centre O with angle theta:
- Draw the line segment [OM]
- Measure angle theta from OM using a protractor, in the specified direction
- Transfer the same length OM onto the new radius: OM’ = OM
Thus, point M’ satisfies two essential conditions: OM’ = OM (same distance from the centre) and angle MOM’ = theta.
Example
Let O be a point and M a point such that OM = 3 cm. We apply a rotation with centre O and angle 90 degrees in the forward direction (anti-clockwise).
| Data | Value |
|---|---|
| Centre | O |
| Angle | 90 degrees |
| OM | 3 cm |
| OM' | 3 cm (remains the same) |
| Angle MOM' | 90 degrees |
Common pitfall
Be careful with the direction of rotation! A 90-degree angle in a clockwise direction does not produce the same image as a 90-degree angle in an anti-clockwise (trigonometric) direction. You must always specify the direction; otherwise, the construction is ambiguous. Furthermore, the centre O is the only invariant point (its own image): do not confuse it with any other point in the figure.