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GCSE Maths revision · Geometry and measures

Properties of shapes

Foundation and Higher

The idea

Each quadrilateral is defined by its sides, angles and diagonals. A rhombus has four equal sides and diagonals that cross at right angles; a parallelogram has two pairs of parallel sides; a kite has one line of symmetry.

  1. List what you are told: equal sides, parallel sides, right angles.
  2. Match those facts to the shape that must have them.
  3. Count lines of symmetry and the order of rotational symmetry separately.

Watch out Use the marks on a diagram, not how it looks: it may not be drawn accurately.

See it worked

Worked exampleNon-calculator1 mark

A rhombus is not a square. What is its order of rotational symmetry?

  1. A non-square rhombus matches itself after 180∘180^\circ and 360∘.360^\circ.
  2. There are two matching positions in a full turn, so its order is 2.

Answer 2

Your turn

Question 1Non-calculator3 marks

This question is about special quadrilaterals.

(a) Write down the name of the quadrilateral that has exactly one pair of parallel sides.

(b) Write down the number of lines of symmetry of a rhombus that is not a square.

(c) Rosa says, "A rectangle is a special kind of rhombus." Explain why Rosa is wrong.

Write your answer, then show the working
  1. (a) A trapezium is defined by having exactly one pair of parallel sides.
  2. (b) The two diagonals of a rhombus are its mirror lines. The lines joining midpoints of opposite sides are not, unless all its angles are right angles.
  3. (c) A rhombus must have four equal sides.
  4. A rectangle only needs four right angles; its length and width can differ, so it is not always a rhombus.

Answer (a) Trapezium; (b) 2; (c) A rhombus has four equal sides, but a rectangle's sides need not all be equal.

Question 2Non-calculator3 marks

A quadrilateral has exactly one pair of parallel sides, and its two non-parallel sides are equal in length.

(a) Which quadrilateral is this? Select one answer.

(b) How many lines of symmetry does it have?

(c) What is its order of rotational symmetry?

(a)
Question 3Non-calculator4 marks

Quadrilateral PQRS has vertices P(0, 0), Q(4, 3), R(9, 3) and S(5, 0).

(a) Show that PQRS is a rhombus.

(b) Work out the area of PQRS.

Write your answer, then show the working
  1. (a) Use Pythagoras for each sloping side. 42+32=5\sqrt{4^{2}+3^{2}}=5
  2. Therefore PQ = √(4² + 3²) = 5, QR = 5, RS = √(4² + 3²) = 5 and SP = 5, so all four sides are equal.
  3. (b) Multiply the base PS by the perpendicular height. 5×35\times 3
  4. Therefore 1515 square units.

Answer (a) PQ = √(4² + 3²) = 5, QR = 5, RS = √(4² + 3²) = 5 and SP = 5, so all four sides are equal.; (b) 1515 square units

In past papers

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