IGCSE · Practice questions · Line and rotational symmetry

Mirror lines and turns.

Six original Cambridge-style questions on symmetry: lines and order of rotational symmetry of 2D shapes, and (Extended) planes of symmetry of solids.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Fold for lines, turn for order.

Lines: test every possible mirror: through vertices, through midpoints of sides.
Order: count the fits in one full turn, including the start; minimum 1.
Angle of turn: \(360^\circ \div \text{order}\).
Regular polygons: \(n\) sides, \(n\) lines, order \(n\).
Solids (Extended): planes along the length from the cross-section's lines, plus any plane across the middle.

Recall

[3 marks]

Write down the number of lines of symmetry and the order of rotational symmetry of (a) a kite (b) a rhombus (c) an isosceles trapezium.

(a) 1 line, order 1 ✓
(b) 2 lines, order 2 ✓
(c) 1 line, order 1 ✓

Recall

[3 marks]

For each capital letter, write down the number of lines of symmetry and the order of rotational symmetry: (a) N (b) X (c) E.

(a) N: 0 lines, order 2 ✓
(b) X: 2 lines, order 2 ✓
(c) E: 1 line (horizontal), order 1 ✓

Calculation

[2 marks]

A regular polygon has rotational symmetry of order 12. Find (a) the smallest angle of turn that maps it onto itself (b) its number of lines of symmetry.

(a) \(360^\circ \div 12 = 30^\circ\) ✓
(b) 12 ✓

Analysis

[2 marks]

A regular polygon maps onto itself after a rotation of \(24^\circ\), and after no smaller rotation. How many sides does it have?

Order \(= 360^\circ \div 24^\circ = 15\). ✓
A regular polygon's order equals its number of sides: 15 sides. ✓

Recall

[4 marks]

(Extended) Write down the number of planes of symmetry of (a) a cube (b) a right pyramid with a square base (c) a cylinder. (d) State the order of rotational symmetry of the square-based pyramid about its vertical axis.

(a) 9 ✓
(b) 4 ✓
(c) infinitely many ✓
(d) order 4 ✓

Analysis

[4 marks]

Triangle PQR has exactly one line of symmetry, which passes through P and meets QR at M. QR = 8 cm and angle QPR = \(52^\circ\).

(a) What type of triangle is PQR? (b) Find angle PQR. (c) Find the length QM. (d) Find angle PMQ, giving a reason.

(a) Isosceles, with PQ = PR. ✓
(b) \((180^\circ - 52^\circ) \div 2 = 64^\circ\). ✓
(c) The mirror line cuts QR in half: QM \(= 8 \div 2 = 4\) cm. ✓
(d) The reflection maps angle PMQ onto angle PMR, so they are equal, and together they make a straight line: each is \(90^\circ\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.