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4024/12 Oct/Nov 2020, Question 22: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 12 (Non-calculator) · Non-calculator · 4 marks

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The question, in short

Circle with centre O through points A, B, C, D, E; angle AOB = 48° and angle DEB = 54°; find the angles x, y and z marked in the diagram.

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

Part (a) · 2 marks

Find x, the angle OAB at A, using the isosceles triangle AOB with angle AOB = 48°.

  1. 1OA = OB (radii), so triangle OAB is isosceles
  2. 2x = (180 − 48) ÷ 2 = 66

Answer66

Part (b) · 1 mark

Find y, the angle BOD at O, using angle DEB = 54° and the circle geometry shown.

  1. 1Angle at centre = 2 × angle at circumference
  2. 2y = 2 × 54 = 108

Answer108

Part (c) · 1 mark

Find z, the angle BCD at C, using the angles already found and the cyclic quadrilateral properties.

  1. 1EBCD is a cyclic quadrilateral, so opposite angles add to 180°
  2. 2z = 180 − 54 = 126

Answer126

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