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4024/11 Oct/Nov 2022, Question 14: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 11 (Non-calculator) · Non-calculator · 5 marks

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The question, in short

In a circle with centre O, tangent AB at B, diameter BD and straight line OCA, angle CDB is x°. Find expressions in x for three angles.

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

Part (a) · 1 mark

Find angle COB in terms of x, in simplest form.

  1. 1Angle COB = 2 × angle CDB (angle at centre is twice angle at circumference) = 2x

Answer2x

Part (b) · 2 marks

Find angle OAB in terms of x, in simplest form.

  1. 1Angle BOC = 2x (angle at centre = 2 × angle at circumference)
  2. 2Angle OBA = 90° (tangent is perpendicular to radius)
  3. 3Angle OAB = 180 − 90 − 2x = 90 − 2x

Answer90 − 2x or 2(45 − x)

Part (c) · 2 marks

Find angle CBO in terms of x, in simplest form.

  1. 1Angle BOC = 2x (angle at centre = 2 × angle at circumference)
  2. 2OB = OC (radii), so angles OBC and OCB are equal
  3. 3Angle CBO = (180 − 2x) ÷ 2 = 90 − x

Answer90 − x

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Also searched as: 4024/11/O/N/22 · 4024_w22_qp_11 · 4024 w22 11 · question 14.
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