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4PM1/01 Jan 2020, Question 6: Perpendicular Lines

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1 · Calculator · 14 marks

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The question, in short

Given points A(3, 0) and B(2, 2), with line L₁ through B perpendicular to AB, find L₁'s equation, then use L₂: x − 7y − 3 = 0 to find midpoint M of AC and the area of triangle ABM.

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

Part (a) · 5 marks

Find an equation of the line L₁ through B perpendicular to AB, giving it in the form ax + by + c = 0.

  1. 1Gradient of AB = (2 − 0)/(2 − 3) = −2
  2. 2Perpendicular gradient = 1/2
  3. 3y − 2 = ½(x − 2)
  4. 4y = ½x + 1
  5. 5x − 2y + 2 = 0

Answerx − 2y + 2 = 0

Part (b) · 5 marks

Find the coordinates of M, the midpoint of AC, where C is where L₁ meets L₂: x − 7y − 3 = 0.

  1. 1From L₁: x = 2y − 2
  2. 2Substitute into L₂: 2y − 2 − 7y − 3 = 0, so −5y = 5
  3. 3y = −1, x = −4, so C = (−4, −1)
  4. 4M = ((3 + (−4))/2, (0 + (−1))/2) = (−1/2, −1/2)

AnswerM = (−1/2, −1/2)

Part (c) · 4 marks

Find the area of the triangle ABM.

  1. 1AB = √((3 − 2)² + (0 − 2)²) = √5
  2. 2BC = √((2 + 4)² + (2 + 1)²) = √45
  3. 3Angle ABC = 90°, so area ABC = ½ × √5 × √45 = 7.5
  4. 4M is the midpoint of AC, so area ABM = ½ × 7.5 = 3.75

Answer3.75

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