## Symmetry

There are two types of symmetry: line symmetry, which involves reflection, and rotational symmetry, which involves rotation. The total order of symmetry of a shape is the sum of the number of lines of symmetry and the order of rotational symmetry of the figure.

### Line Symmetry

A figure has a line of symmetry if it maps onto itself under reflection in the line.

e.g.

• A rectangle has 2 axes of symmetry. (m and n are axes of symmetry.)
• A regular hexagon has 6 axes of symmetry.
• A circle has an infinite number of axes of symmetry.

• The figure below has no axis of symmetry.

### Rotational Symmetry

A figure has rotational symmetry if it maps onto itself under rotation about a point at its centre.

The order of rotational symmetry is the number of times the shape maps onto itself during a rotation of 360°.

e.g.

• A rectangle has order of rotational symmetry of 2.

180° and 360° rotations will map it onto itself.

• A regular hexagon has order of rotational symmetry of 6.
• A scalene triangle, with no equal sides or angles has

order of rotational symmetry of 1.

• All figures have an order of rotational symmetry of at least 1.

### Total Order of Symmetry

The total order of symmetry = number of axes of symmetry + order of rotational symmetry.

The table shows the symmetry properties of some common shapes.

 Shape Axes of symmetry Order of rotational symmetry Total order of symmetry Scalene triangle 0 1 1 Isosceles triangle 1 1 2 Equilateral triangle 3 3 6 Kite 1 1 2 Trapezium 0 1 1 Isosceles trapezium 1 1 2 Parallelogram 0 2 2 Rhombus 2 2 4 Rectangle 2 2 4 Square 4 4 8 Regular pentagon 5 5 10 Regular hexagon 6 6 12 Regular octagon 8 8 16

A figure has point symmetry if it maps onto itself under a rotation of 180° (a half turn).

### Summary of Transformations

 Reflection Rotation Translation Enlargement Length, angle size and area are invariant Length, angle size and area are invariant Length, angle size and area are invariant Angle size invariant Indirect Direct Direct Direct Isometry Isometry Isometry Not isometry Points on mirror line are invariant Centre of rotation is invariant No invariant points Centre of enlargement is invariant