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4024/22 Oct/Nov 2024, Question 8: Similar Triangles

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 12 marks

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The question, in short

Using the diagram with DE parallel to BC and given lengths BD = 2.7, DE = 5.6, CE = 3.9, BC = 9.8, work through similarity, side lengths, an angle and an area.

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

Part (a) · 3 marks

Fill in the missing angle names and give the reason for each pair, completing the similarity argument for triangles ABC and ADE.

  1. 1Angle BAC = angle DAE, because common angle
  2. 2Angle ABC = angle ADE, because corresponding angles (DE parallel to BC)
  3. 3Angle ACB = angle AED, because corresponding angles

Answerangle BAC = angle DAE angle ABC = angle ADE, corresponding angles angle ACB = angle AED, corresponding angles

Part (b) · 4 marks

Use similarity to work out the lengths AD and AE, showing that AD = 3.6 cm and AE = 5.2 cm.

  1. 1AD/AB = DE/BC: AD/(AD + 2.7) = 5.6/9.8
  2. 29.8AD = 5.6AD + 15.12, so 4.2AD = 15.12, AD = 3.6 cm
  3. 3AE/(AE + 3.9) = 5.6/9.8: 9.8AE = 5.6AE + 21.84
  4. 44.2AE = 21.84, AE = 5.2 cm

AnswerAD/(AD + 2.7) = 5.6/9.8 and AE/(AE + 3.9) = 5.6/9.8, correct elimination of fraction and rearrangement leading to AD = 3.6 and AE = 5.2

Part (c) · 3 marks

Find the size of angle DAE.

  1. 1cos DAE = (3.6² + 5.2² − 5.6²)/(2 × 3.6 × 5.2)
  2. 2= 8.64/37.44 = 0.2307…
  3. 3Angle DAE = 76.65…° ≈ 76.7°

Answer76.7 or 76.65 to 76.66

Part (d) · 2 marks

Given AD = 3.6 cm, AE = 5.2 cm and angle DAE = 76.65°, work out the area of triangle ABC.

  1. 1Use AD = 3.6 and AE = 5.2 from part (b), and angle A = 76.65…° from part (c)
  2. 2AB = 3.6 + 2.7 = 6.3 cm and AC = 5.2 + 3.9 = 9.1 cm
  3. 3Area = 1/2 × 6.3 × 9.1 × sin 76.65…°
  4. 4= 27.89… ≈ 27.9 cm²

Answer27.9 or 27.87 to 27.90

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