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0580/13 May/Jun 2025, Question 18: Combining Circle Theorems

Cambridge IGCSE Mathematics 0580 · Paper 13 (Core) · Non-calculator · 5 marks

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The question, in short

Using the circle with centre O, points P and S on it, POR straight and tangent QRST at S, find x with a reason and then find y.

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

Part (a) · 2 marks

Find x, the angle OSP, and state a geometrical reason for your value.

  1. 1OP = OS (radii), so triangle OPS is isosceles
  2. 2x = 25, because base angles of an isosceles triangle are equal

Answer25, base angles of an isosceles triangle are equal

Part (b) · 3 marks

Find y, the angle QRO, using the circle and tangent properties shown.

  1. 1Angle OSR = 90° (tangent is perpendicular to radius)
  2. 2Angle ROS = 25 + 25 = 50° (exterior angle of triangle OPS)
  3. 3Angle ORS = 180 − 90 − 50 = 40°
  4. 4y = 180 − 40 = 140 (angles on a straight line)

Answer140

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Also searched as: 0580/13/M/J/25 · 0580_s25_qp_13 · 0580 s25 13 · question 18.
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