IGCSE Maths / Transformation and similarity

Lines of symmetry and rotational symmetry

Edexcel International GCSE Maths (4MA1), chapter 14: Transformation and similarity, specification point 4.3a. Free lesson.

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A line of symmetry is a fold line that maps a shape onto itself. The order of rotational symmetry counts how many times a shape matches itself during a full turn, so these two answers must be worked out separately.

What counts as a line of symmetry

A line of symmetry is a fold line. When you fold the shape along it, the two halves must land exactly on top of each other. Halves that are merely the same size do not make a line of symmetry. A square has 4 lines of symmetry: 2 through opposite corners and 2 through the midpoints of opposite sides. A rectangle has 2 lines of symmetry. Its diagonals do not work as fold lines. A rhombus has 2 lines of symmetry: its diagonals. A regular hexagon has 6 lines of symmetry.

How to find the order of rotational symmetry

The order of rotational symmetry is the number of times a shape looks exactly the same as it turns through one full turn about its centre. The full turn back to the start counts, so the smallest possible order is 1, never 0. A square matches itself 4 times in one full turn, so its order is 4. A rectangle has order 2. A parallelogram also has order 2, even though it has no lines of symmetry. For a regular shape with n sides, there are n lines of symmetry and its order is n.

Why the two symmetry answers differ

Lines of symmetry and rotational order ask different questions. A parallelogram shows why: turning it halfway round puts it back on itself, but folding it along either diagonal does not make the halves land on top of each other. A parallelogram has 0 lines of symmetry and rotational order 2. Its diagonals cut each other in half, but that does not make them fold lines. To check a possible fold line, follow a corner through the fold. If it does not land on another corner exactly, the line is not a line of symmetry.

Use a fold line to work out angles

A fold line can give you equal angles. If folding swaps two parts, the angles in those parts are equal. When the fold line runs from a corner to the opposite side, it meets that side at a right angle. In triangle ABC, AB = AC and AM is the line of symmetry. Folding along AM swaps B and C, so the angles at B and C are equal. The fold also swaps the two angles at M. They lie on the straight line BC, so they add to 180°. Since they are equal, each is 180° ÷ 2 = 90°. Therefore, angle AMC = 90°.

Check the regular pentagon

For a regular shape, the number of lines of symmetry and the rotational order both match the number of sides. A regular pentagon has 5 equal sides and 5 equal angles. Lines of symmetry: 5. Order of rotational symmetry: 5.

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