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0580/22 Oct/Nov 2025, Question 17: Length of a Line Segment

Cambridge IGCSE Mathematics 0580 · Paper 22 (Extended) · Non-calculator · 11 marks

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The question, in short

Kite ABCD has B at (−3, 1) and D at (−5, 9), with BD a diagonal. Use the diagonal ratio and midpoint to find AC's length and equation.

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

Part (a) · 5 marks

Given diagonals BD : AC = 2 : 3, calculate AC's length, giving a simplified surd.

  1. 1BD = √((−5 − (−3))² + (9 − 1)²) = √(4 + 64) = √68
  2. 2√68 = √(4 × 17) = 2√17
  3. 3AC = 3/2 × BD = 3/2 × 2√17
  4. 4AC = 3√17

Answer3√17

Part (b) · 2 marks

Determine the midpoint coordinates of the segment joining B and D.

  1. 1x: (−3 + (−5)) ÷ 2 = −4; y: (1 + 9) ÷ 2 = 5
  2. 2Midpoint = (−4, 5)

Answer(−4, 5)

Part (c) · 4 marks

Using that AC passes through BD's midpoint, find AC's equation as y = mx + c.

  1. 1Gradient BD = (9 − 1) ÷ (−5 − (−3)) = 8 ÷ (−2) = −4
  2. 2Kite diagonals are perpendicular, so gradient AC = 1/4
  3. 3Midpoint of BD = ((−3 − 5)/2, (1 + 9)/2) = (−4, 5)
  4. 45 = 1/4 × (−4) + c, so c = 6
  5. 5y = (1/4)x + 6

Answery = (1/4)x + 6

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Also searched as: 0580/22/O/N/25 · 0580_w25_qp_22 · 0580 w25 22 · question 17.
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