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4MA1/2HR Jun 2023, Question 25: The Perpendicular Bisector

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2HR (Higher) · Calculator · 6 marks

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The question, in short

Given line y − 2x = 7 is the perpendicular bisector of AB with A(j, 7) and B(6, k), find the midpoint coordinates of AB, showing algebraic working.

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

Solution · 6 marks

  1. 1y − 2x = 7 rearranges to y = 2x + 7, so the bisector has gradient 2
  2. 2AB is perpendicular to the bisector, so AB has gradient −1/2
  3. 3Gradient of AB = (k − 7)/(6 − j) = −1/2, so 2k − 14 = j − 6 and j = 2k − 8
  4. 4The midpoint ((j + 6)/2, (7 + k)/2) lies on the bisector
  5. 5So (7 + k)/2 = 2 × (j + 6)/2 + 7, which gives 7 + k = 2j + 26 and k = 2j + 19
  6. 6Substitute j = 2k − 8: k = 2(2k − 8) + 19 = 4k + 3, so 3k = −3 and k = −1
  7. 7Then j = 2 × (−1) − 8 = −10
  8. 8Midpoint = ((−10 + 6)/2, (7 + (−1))/2) = (−2, 3)

Answer(−2, 3)

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