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0580/43 Oct/Nov 2023, Question 9: Length of a Line Segment

Cambridge IGCSE Mathematics 0580 · Paper 43 (Extended) · Calculator · 14 marks

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The question, in short

Given points A(0, 2), B(3, 3) and C(4, 0), work with triangle ABC and a kite ABCD, finding side lengths, line equations and the coordinates of D.

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

Part (a) · 4 marks

Work out the three side lengths of triangle ABC and use them to decide whether it is scalene, isosceles or equilateral, showing all working.

  1. 1AB² = (3 − 0)² + (3 − 2)² = 9 + 1 = 10
  2. 2AC² = (4 − 0)² + (0 − 2)² = 16 + 4 = 20
  3. 3BC² = (4 − 3)² + (0 − 3)² = 1 + 9 = 10
  4. 4AB = BC ≠ AC, so triangle ABC is isosceles

AnswerIsosceles

Part (bi) · 3 marks

Work out the equation of the line through A and C, giving it as y = mx + c.

  1. 1Gradient AC = (0 − 2)/(4 − 0) = −1/2
  2. 2y-intercept is 2 (A is (0, 2))
  3. 3y = −1/2 x + 2

Answery = −1/2 x + 2

Part (bii) · 4 marks

Work out the equation of the line that cuts AC in half at right angles, giving it as y = mx + c.

  1. 1Midpoint of AC = ((0 + 4)/2, (2 + 0)/2) = (2, 1)
  2. 2Perpendicular gradient = −1 ÷ (−1/2) = 2
  3. 31 = 2 × 2 + c → c = −3
  4. 4y = 2x − 3

Answery = 2x − 3

Part (biii) · 3 marks

Given ABCD is a kite and D is (w, 4w + 1), work out the coordinates of D.

  1. 1In kite ABCD, D lies on the perpendicular bisector of AC (like B), so 4w + 1 = 2w − 3
  2. 22w = −4 → w = −2
  3. 3D = (−2, 4 × (−2) + 1) = (−2, −7)

Answer(−2, −7)

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Also searched as: 0580/43/O/N/23 · 0580_w23_qp_43 · 0580 w23 43 · question 9.
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