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4PM1/02 Nov 2021, Question 9: Trigonometry in 3D: Pyramids and Cones

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 2 · Calculator · 10 marks

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The question, in short

Triangular pyramid ABCD has base ABC with AB = BC = 10 cm, ∠ABC = 90°, and AD = CD = 8 cm. Work out lengths, an angle between planes, and a vertical height.

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

Part (a) · 2 marks

Calculate AC exactly, giving it as p√q cm with p an integer and q prime.

  1. 1AC² = 10² + 10² = 200
  2. 2AC = 10√2 cm

AnswerAC = 10√2 cm

Part (b) · 2 marks

M is the midpoint of AC. Calculate BM exactly as m√n cm, with m and n both prime.

  1. 1AM = 5√2 and BM ⟂ AC (isosceles triangle)
  2. 2BM = √(10² − (5√2)²) = √50 = 5√2 cm

AnswerBM = 5√2 cm

Part (c) · 4 marks

Given BD = 6 cm, calculate the acute angle between planes ACD and ABC, in degrees to one decimal place.

  1. 1DM ⟂ AC: DM = √(8² − (5√2)²) = √14
  2. 2Required angle is DMB
  3. 3cos DMB = (14 + 50 − 36)/(2 × √14 × 5√2)
  4. 4∠DMB = 58.1°

Answer58.1°

Part (d) · 2 marks

With base ABC horizontal, calculate how far D is vertically above the base, in cm to 3 significant figures.

  1. 1Height = DM sin(∠DMB)
  2. 2= √14 × sin 58.05…°
  3. 3= 3.17 cm

Answer3.17 cm (accept 3.18 cm)

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