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0580/21 May/Jun 2025, Question 10: Angle at the Centre

Cambridge IGCSE Mathematics 0580 · Paper 21 (Extended) · Non-calculator · 4 marks

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The question, in short

Using a circle with centre O, tangent DE at A, and given angles ACO = 35° and BCO = 40°, find four angles involving the triangle and tangent.

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Part (a) · 1 mark

Find angle AOC using triangle AOC and the given angle at C.

  1. 1OA = OC (radii), so angle OAC = 35°
  2. 2Angle AOC = 180 − 35 − 35 = 110°

Answer110

Part (b) · 1 mark

Find angle ABC, the angle at B subtended by chord AC.

  1. 1Angle at centre is twice angle at circumference
  2. 2Angle ABC = 110 ÷ 2 = 55°

Answer55

Part (c) · 1 mark

Find angle DAC between tangent DA and chord AC.

  1. 1Alternate segment theorem: angle DAC = angle ABC
  2. 2Angle DAC = 55°

Answer55

Part (d) · 1 mark

Find angle OAB in triangle OAB.

  1. 1OB = OC (radii), so angle OBC = angle OCB = 40°
  2. 2Angle OBA = 55 − 40 = 15°
  3. 3OA = OB, so angle OAB = angle OBA = 15°

Answer15

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Also searched as: 0580/21/M/J/25 · 0580_s25_qp_21 · 0580 s25 21 · question 10.
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