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4PM1/01R Jun 2024, Question 3: Area of a Triangle Using Sine

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 8 marks

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The question, in short

Triangle ABC has AB = x cm, AC = (x + 7) cm, angle BAC = 30° and area 36 cm². Show x = 9, then find BC and the other two angles.

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

Part (a) · 3 marks

Use the area formula with sides x and x + 7 and included angle 30° to form an equation, then show it gives x = 9.

  1. 136 = ½ × x × (x + 7) × sin 30°
  2. 236 = x(x + 7)/4, so x² + 7x − 144 = 0
  3. 3(x − 9)(x + 16) = 0
  4. 4x = 9 (x = −16 is rejected)

Answerx = 9 shown

Part (b) · 2 marks

Using x = 9, calculate the length of BC in cm to 3 significant figures.

  1. 1BC² = 9² + 16² − 2 × 9 × 16 × cos 30°
  2. 2BC = 9.3586… ≈ 9.36

AnswerBC = 9.36 cm

Part (c) · 3 marks

Find angle ABC and angle ACB in degrees to one decimal place.

  1. 1sin ABC/16 = sin 30°/9.3586…
  2. 2The sine rule gives 58.74°, but ABC is opposite the longest side, so it is obtuse
  3. 3∠ABC = 180° − 58.74° = 121.3°
  4. 4∠ACB = 180° − 30° − 121.26° = 28.7°

Answer(i) ∠ABC = 121.3° (or 121.2°) (ii) ∠ACB = 28.7° (or 28.8°)

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