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4024/12 May/Jun 2026, Question 16: Proving Triangles Are Similar

Cambridge O Level Mathematics 4024 · Paper 12 (Non-calculator) · Non-calculator · 6 marks

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The question, in short

Show two triangles in a rhombus are similar using angle reasons, then use the midpoint and given lengths to find a length and an area ratio.

Summarised in our own words. Use the official paper for the exact wording and the diagram.

Part (a) · 3 marks

Prove triangle AXB is similar to triangle MXD, justifying each statement with a geometrical reason.

  1. 1Angle AXB = angle MXD (vertically opposite angles)
  2. 2AB is parallel to DC, so angle BAX = angle DMX (alternate angles)
  3. 3Angle ABX = angle MDX (alternate angles)
  4. 4All three angles are equal, so triangle AXB is similar to triangle MXD

AnswerAngle AXB = angle DXM (vertically opposite angles); angle BAX = angle DMX (alternate angles); angle ABX = angle XDM (alternate angles); all angles equal, so the triangles are similar

Part (bi) · 2 marks

Given AB = 8 cm and BD = 15 cm with M the midpoint of DC, calculate DX.

  1. 1DM = 8 ÷ 2 = 4, so scale factor AXB to MXD is 8 : 4 = 2 : 1
  2. 2BX : XD = 2 : 1
  3. 3DX = 15 ÷ 3 = 5 cm

Answer5

Part (bii) · 1 mark

Complete the ratio of the area of triangle AXB to the area of triangle MXD in the form … : 1.

  1. 1Length ratio 2 : 1
  2. 2Area ratio = 2² : 1² = 4 : 1

Answer4

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