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4024/21 Oct/Nov 2024, Question 9: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 10 marks

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The question, in short

Use the given triangle and circle diagrams to work out unknown angles and a triangle's area.

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

Part (a) · 4 marks

Using parallel lines and equal sides in the diagram, determine the sizes of angles x and y.

  1. 1Angle ABE = angle BEC = 28° (alternate angles, AB parallel to EC)
  2. 2x = 180 − 54 − 28 = 98 (angles in a triangle add to 180°)
  3. 3Angle CED = angle BAE = 54° (corresponding angles)
  4. 4CD = DE so angle ECD = angle CED = 54° (isosceles triangle)
  5. 5y = 180 − 54 − 54 = 72

Answerx = 98, y = 72

Part (b) · 6 marks

With tangent QR = 12 cm and angle RPQ = 35°, work out the area of triangle PQR.

  1. 1Angle ORQ = 90° (tangent meets radius at right angles)
  2. 2OP = OR, so angle ORP = 35°; angle ROQ = 35 + 35 = 70°, so angle OQR = 20°
  3. 3OR = 12 tan 20° = 4.367…, OQ = 12 ÷ cos 20° = 12.770…, so PQ = 17.137…
  4. 4Area = 1/2 × 12 × 17.137… × sin 20°
  5. 5= 35.16… ≈ 35.2 cm²

Answer35.2 or 35.16 to 35.24

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