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0580/42 Feb/Mar 2024, Question 2: Finding Missing Sides

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 8 marks

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The question, in short

Circle with points A, B, C, D; lines ADX and BCX extended meet at X; angle BAD = x° and angle DCX = y°. Explain the angle equality, prove two triangles similar, then use AD = 15 cm, DX = 9 cm, CX = 12 cm to find BC and an area ratio.

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

Part (a) · 2 marks

Give a geometrical reason for each step to justify why angle x equals angle y.

  1. 1y + angle BCD = 180° (angles on a straight line)
  2. 2x + angle BCD = 180° (opposite angles of a cyclic quadrilateral are supplementary)
  3. 3So x = y

Answery + angle BCD = 180 (angles on a straight line) and x + angle BCD = 180 (opposite angles of a cyclic quadrilateral are supplementary, or angles in opposite segments are supplementary), so x = y

Part (b) · 2 marks

Demonstrate that triangle ABX and triangle CDX are similar.

  1. 1Angle AXB = angle CXD (common angle)
  2. 2Angle BAX = angle DCX (x = y from part (a))
  3. 3So angle ABX = angle CDX (angles in a triangle add to 180°)
  4. 4All three angles are equal, so triangles ABX and CDX are similar

AnswerAngle AXB = angle CXD (common angle), angle BAX = angle DCX (x = y) and angle ABX = angle CDX, so all angles are equal and the triangles are similar

Part (ci) · 3 marks

Using AD = 15 cm, DX = 9 cm and CX = 12 cm, work out the length BC.

  1. 1AX = 15 + 9 = 24 cm; A corresponds to C, B to D
  2. 2Scale factor = AX/CX = 24/12 = 2
  3. 3BX = 2 × DX = 2 × 9 = 18 cm
  4. 4BC = 18 − 12 = 6 cm

Answer6

Part (cii) · 1 mark

Fill in the missing value in the ratio of the area of triangle ABX to the area of triangle CDX, given as …… : 1.

  1. 1Length scale factor = 2, so area scale factor = 2² = 4
  2. 2Ratio = 4 : 1

Answer4

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Also searched as: 0580/42/F/M/24 · 0580_m24_qp_42 · 0580 m24 42 · question 2.
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