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0580/21 Oct/Nov 2025, Question 14: Combining Circle Theorems

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Non-calculator · 9 marks

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The question, in short

Circle with points A, B, C, D, E; chords AC and BD meet at X; angle ACD = 55° and angle CXD = 88°. Find angles with reasons, then use similar triangles CXD and BXA with DX = 8.0 cm, BX = 2.7 cm, AX = 4.0 cm.

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Part (a) · 6 marks

Find angle CDB, angle ABD and angle AED, giving a geometrical reason for each value.

  1. 1Angle CDB = 180 − 55 − 88 = 37°, because the angle sum of a triangle is 180°
  2. 2Angle ABD = angle ACD = 55°, because angles in the same segment are equal
  3. 3Angle AED = 180 − 55 = 125°, because opposite angles of a cyclic quadrilateral (ACDE) add to 180°

AnswerAngle CDB = 37, angle sum of triangle = 180 Angle ABD = 55, angles in the same segment are equal Angle AED = 125, opposite angles of cyclic quadrilateral sum to 180

Part (bi) · 2 marks

Using the similar triangles CXD and BXA, work out the length of CX in cm.

  1. 1Corresponding sides: CX ↔ BX, DX ↔ AX
  2. 2CX/2.7 = 8.0/4.0 = 2
  3. 3CX = 2 × 2.7 = 5.4 cm

Answer5.4

Part (bii) · 1 mark

Complete the ratio of the area of triangle CXD to the area of triangle BXA.

  1. 1Length scale factor = 8.0 ÷ 4.0 = 2
  2. 2Area scale factor = 2² = 4
  3. 3Area CXD : area BXA = 4 : 1

Answer4 : 1

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