Discovering axial symmetry

What is axial symmetry?

Axial symmetry (or orthogonal symmetry) transforms a shape into its ‘reflection’ across a straight line, known as the axis of symmetry. This is exactly what happens with a mirror: the reflection of an object is its symmetrical image across the plane of the mirror.

Example: the letters A, H, M, O, T, U, V, W, X and Y have a vertical axis of symmetry. A butterfly, a human face or a leaf often have an axis of symmetry.

The symmetric of a point

Let (d) be a line and A a point. The symmetric of A with respect to (d), denoted by A’, is the point such that (d) is the perpendicular bisector of the segment [AA’].

This means two things:

• (d) intersects [AA'] perpendicularly,
• (d) intersects [AA'] at its midpoint.

If A is already on (d), then A' = A: a point on the line is its own symmetric image.

Properties preserved

Property	Preserved by axial symmetry?
Lengths	Yes
Angles	Yes
Alignment of points	Yes
Direction of rotation	No (the figure is inverted)

Common pitfall

Note: the axis of symmetry is not necessarily horizontal or vertical; it may be oblique. You must always check that the line is indeed perpendicular to the segment [AA'] AND passes through its midpoint; otherwise, it is not a valid axial symmetry.