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4024/11 Oct/Nov 2023, Question 15: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 11 (Non-calculator) · Non-calculator · 4 marks

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The question, in short

Circle centre O with points B, C, D on it and tangents AB, AC from A, angle BAC = 38°. Find angles ABC, BOC and BDC.

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

Part (a) · 1 mark

Find angle ABC, using the tangent lengths from A and the given angle at A.

  1. 1AB = AC (tangents from the same point), so triangle ABC is isosceles
  2. 2Angle ABC = (180 − 38) ÷ 2 = 71°

Answer71

Part (b) · 2 marks

Find angle BOC, using the tangent radii and the angle at A.

  1. 1Angle ABO = angle ACO = 90° (tangent meets radius at 90°)
  2. 2Angle BOC = 360 − 90 − 90 − 38 = 142°

Answer142

Part (c) · 1 mark

Find angle BDC, using the angle at the centre and the point D on the circle.

  1. 1Tangent meets radius at 90°: angles OBA = OCA = 90°
  2. 2Angle BOC = 360 − 90 − 90 − 38 = 142°
  3. 3Angle at the centre is twice the angle at the circumference
  4. 4Angle BDC = 142 ÷ 2 = 71°

Answer71

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Also searched as: 4024/11/O/N/23 · 4024_w23_qp_11 · 4024 w23 11 · question 15.
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