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4PM1/01R Jun 2023, Question 3: Area of a Triangle Using Sine

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 9 marks

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The question, in short

Use the cosine rule on triangle ABC with AB = 10 cm, AC = 8 cm and angle BAC = 100° to work out side x, the other angles and the area of triangle BMC.

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

Part (a) · 2 marks

Calculate BC = x using the cosine rule with two sides 10 cm and 8 cm and included angle 100°, giving 3 significant figures.

  1. 1x² = 10² + 8² − 2 × 10 × 8 × cos 100°
  2. 2x² = 164 + 27.78… = 191.78…
  3. 3x = 13.848… ≈ 13.8

Answerx = 13.8

Part (b) · 3 marks

Work out angle ABC and angle ACB in degrees to one decimal place, using the sine rule or cosine rule.

  1. 1sin ABC = 8 sin 100°/13.848…
  2. 2ABC = 34.675…° ≈ 34.7°
  3. 3ACB = 180° − 100° − 34.675…° = 45.3°

Answer(i) 34.7° (ii) 45.3°

Part (c) · 4 marks

With BM bisecting angle ABC and meeting AC at M, calculate the area of triangle BMC in cm² to 3 significant figures.

  1. 1Angle MBC = 34.675°/2 = 17.337°
  2. 2Angle BMC = 180° − 17.337° − 45.324° = 117.34°
  3. 3MC = 13.848 sin 17.337°/sin 117.34° = 4.645…
  4. 4Area BMC = (1/2) × 4.645 × 13.848 × sin 45.324°
  5. 5= 22.87… ≈ 22.9 cm²

Answer22.9 cm² (accept 22.8)

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