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4MA1/2H Jan 2023, Question 14: Combining Circle Theorems

Pearson Edexcel International GCSE Mathematics A 4MA1 · Paper 2H (Higher) · Calculator · 5 marks

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The question, in short

For a cyclic quadrilateral ABCD on a circle centre O with tangent EBF at B, angle DAB = 40° and angle ABF = 66°, find angle DCB with a reason, then angle ADO.

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

Part (ai) · 1 mark

Find the size of angle DCB.

  1. 1ADCB is a cyclic quadrilateral, so angle DAB and angle DCB add up to 180°
  2. 2180 − 40 = 140, so angle DCB = 140°

Answer140

Part (aii) · 1 mark

State the circle theorem that justifies the value of angle DCB.

  1. 1A, D, C and B all lie on the circle, so ADCB is a cyclic quadrilateral

AnswerOpposite angles of a cyclic quadrilateral add up to 180°

Part (b) · 3 marks

Find the size of angle ADO using the tangent and radii properties.

  1. 1The angle at the centre is twice the angle at the circumference: angle DOB = 2 × 40 = 80°
  2. 2OD and OB are radii, so triangle DOB is isosceles
  3. 3Angle ODB = (180 − 80) ÷ 2 = 50°
  4. 4By the alternate segment theorem, angle ADB = angle ABF = 66°
  5. 5Angle ADO = angle ADB − angle ODB = 66 − 50 = 16°

Answer16

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