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4024/22 May/Jun 2023, Question 3: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 8 marks

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The question, in short

Use an equilateral triangle joined to a regular polygon with interior angle 165° to find an angle and the polygon's side count, then work with circle geometry involving angle ABC = x°.

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

Part (ai) · 1 mark

Using the 165° interior angle and the equilateral triangle, work out the size of angle y.

  1. 1Angles round a point: y = 360 − 60 − 165
  2. 2y = 135

Answer135

Part (aii) · 2 marks

Determine how many sides the regular polygon A has, using its interior angle of 165°.

  1. 1Exterior angle = 180 − 165 = 15°
  2. 2Number of sides = 360 ÷ 15 = 24

Answer24

Part (bi) · 3 marks

Prove that angle OAC equals (90 − x)°, giving a reason at each step.

  1. 1Angle AOC = 2x (angle at centre is twice angle at circumference)
  2. 2OA = OC (radii), so triangle AOC is isosceles
  3. 3Angle OAC = (180 − 2x)/2 = (90 − x)°

AnswerAngle AOC = 2x (angle at centre is twice angle at circumference); angle OAC = (180 − 2x)/2 = 90 − x (isosceles triangle)

Part (bii) · 2 marks

Given angle BAO = 54° and angle OCB = 11°, calculate the value of x.

  1. 1OA = OB (radii), so angle ABO = angle BAO = 54°
  2. 2OB = OC (radii), so angle OBC = angle OCB = 11°
  3. 3x = angle ABO − angle OBC = 54 − 11
  4. 4x = 43

Answer43

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