Properties of rotation

A transformation that preserves distances

Rotation is what is known as an isometry: it preserves lengths. If M' and N' are the images of M and N under a rotation, then M'N' = MN.

It also preserves:

• angles (an angle of 50 degrees remains an angle of 50 degrees after rotation)
• collinearity (three collinear points have collinear images)
• the areas of figures
• the nature of figures: an equilateral triangle is mapped to an equilateral triangle; a circle is mapped to a circle of the same radius

The special case of a 180-degree angle

A rotation of 180 degrees, regardless of the direction chosen, is called central symmetry with centre O. Indeed, rotating by 180 degrees in one direction or the other amounts to the same thing: point M’ is the point symmetric to M with respect to O, so O is the midpoint of [MM’].

Rotation and circles

If a point M lies on a circle with centre O and radius r, its image M’ under a rotation about centre O remains on the same circle: OM’ = OM = r. This is a very useful property for constructing figures with circular symmetry (regular polygons, for example).

Example of composition

If we apply two rotations with the same centre O, with angles θ₁ and then θ₂, the overall effect is a rotation with centre O and angle (θ₁ + θ₂). For example, a rotation of 40 degrees followed by a rotation of 70 degrees, with the same centre and in the same direction, is equivalent to a rotation of 110 degrees.

Common pitfall

Do not confuse ‘preserves the angles of the figure’ with ‘the angle of rotation remains the same everywhere’: the rotation angle theta is the same for ALL transformed points, but the internal angles of the figure (between two sides of a polygon, for example) are also preserved; these are two different concepts that must not be confused.