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TMUA 2023 · Paper 1 · Question 16 of 20

TMUA 2023 Paper 1 Question 16

Coordinate geometry — Coordinate geometry · right angles at each vertex. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Coordinate geometryCoordinate geometry · right angles at each vertex7 options
A right-angled triangle has vertices at (2,  3), (9,  −1) and (5,  k).

Find the sum of all the possible values of k.

  1. A−8
  2. B−6
  3. C0.25
  4. D2
  5. E2.25
  6. F8.25
  7. G10.25
Show the answer and worked solution
answer · E
  1. A−8
  2. B−6
  3. C0.25
  4. D2
  5. E2.25
  6. F8.25
  7. G10.25
Call the points A(2, 3), B(9, −1) and C(5, k). The right angle may be at any vertex. At A: AB⋅AC = 7(3) + (−4)(k−3) = 0 gives k = 334 = 8.25. At B: BA⋅BC = (−7)(−4) + 4(k+1) = 0 gives k = −8. At C: CA⋅CB = (−3)(4) + (3−k)(−1−k) = k2 − 2k − 15 = 0 gives k = 5 or k = −3. The sum is 8.25 − 8 + 5 − 3 = 2.25.