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4PM1/01R Jun 2022, Question 9: Vector Geometry

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 10 marks

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The question, in short

Quadrilateral ABCD has AD = 2a + b, BC = (1/3)b and BD = −4a − b. Show AB is parallel to DC, then use vectors to find AY where diagonals AC and BD meet at Y.

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

Part (a) · 4 marks

Prove that side AB is parallel to side DC using the given vectors.

  1. 1AB = AD + DB = 2a + b − (−4a − b) = 6a + 2b = 2(3a + b)
  2. 2DC = DB + BC = 4a + b + (1/3)b = 4a + (4/3)b = (4/3)(3a + b)
  3. 3AB = (3/2)DC, a multiple of DC
  4. 4So AB is parallel to DC

AnswerAB = 6a + 2b = 2(3a + b), DC = (4/3)(3a + b), so AB = (3/2)DC and AB is parallel to DC

Part (b) · 6 marks

Using a vector method, express AY in simplified form in terms of a and b, where Y is the intersection of AC and BD.

  1. 1AC = AB + BC = 6a + 2b + (1/3)b = 6a + (7/3)b
  2. 2AY = λAC = λ(6a + (7/3)b)
  3. 3AY = AD + μDB = 2a + b + μ(4a + b)
  4. 4a: 2 + 4μ = 6λ; b: 1 + μ = (7/3)λ
  5. 5λ = 3/5 (μ = 2/5)
  6. 6AY = (3/5)(6a + (7/3)b) = (18/5)a + (7/5)b

AnswerAY = (18/5)a + (7/5)b

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