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4024/11 May/Jun 2024, Question 16: Combining Circle Theorems

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

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The question, in short

Circle with centre O and points A, B, C, D on circumference; angle BCD = 58° and angle DBC = 72°. Find angle DAB with reason, angle DOC, and angle BCO.

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Part (a) · 2 marks

Complete the statement giving the size of angle DAB and the reason for it.

  1. 1Angle DAB = 180 − 58 = 122°
  2. 2because opposite angles of a cyclic quadrilateral add to 180° (angles in opposite segments are supplementary)

AnswerAngle DAB = 122° because angles in opposite segments are supplementary

Part (bi) · 1 mark

Find angle DOC.

  1. 1Angle DOC = 2 × angle DBC = 2 × 72 = 144° (angle at centre is twice angle at circumference)

Answer144

Part (bii) · 2 marks

Find angle BCO.

  1. 1Angle at centre = 2 × angle at circumference: DOC = 2 × 72 = 144°
  2. 2OD = OC (radii), so angle OCD = (180 − 144) ÷ 2 = 18°
  3. 3Angle BCO = 58 − 18 = 40°

Answer40

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