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4PM1/02R Jun 2025, Question 2: The Point Dividing a Line in a Ratio

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

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The question, in short

Given P(−7, 3) and Q(3, 18), with R dividing PQ in ratio 2 : 3, find the perpendicular line through R and the area of triangle QRS where l meets the x-axis at S.

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Part (a) · 5 marks

Find the equation of line l through R perpendicular to PQ, giving it as ay + bx + c = 0 with integer coefficients.

  1. 1R = (−7 + (2/5)(10), 3 + (2/5)(15)) = (−3, 9)
  2. 2Gradient of PQ = (18 − 3)/(3 − (−7)) = 3/2
  3. 3Gradient of l = −2/3
  4. 4y − 9 = −(2/3)(x + 3)
  5. 53y + 2x − 21 = 0

Answer3y + 2x − 21 = 0

Part (b) · 5 marks

Line l meets the x-axis at S; find the area of triangle QRS.

  1. 1y = 0: 2x − 21 = 0, so S = (10.5, 0)
  2. 2QR = √((3 + 3)² + (18 − 9)²) = 3√13
  3. 3RS = √((10.5 + 3)² + (0 − 9)²) = 9√13/2
  4. 4Angle QRS = 90°, so area = (1/2) × 3√13 × 9√13/2
  5. 5Area = 351/4

Answer351/4 (= 87.75)

Stuck on a step? Read the free lesson: The Point Dividing a Line in a Ratio

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