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4024/21 Oct/Nov 2023, Question 10: Circumference and Area of a Circle

Cambridge O Level Mathematics 4024 · Paper 21 (Calculator) · Calculator · 8 marks

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The question, in short

For a circle centred at O with diameters AC and BD, prove two triangles congruent, then use diameter 10 cm and AB = 9 cm to find how much larger the circle's circumference is than the shaded shape's perimeter.

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

Part (a) · 3 marks

Prove that triangle OAB and triangle ODC are congruent, justifying every statement you write down.

  1. 1AO = DO (radii of the same circle)
  2. 2BO = CO (radii of the same circle)
  3. 3Angle AOB = angle DOC (vertically opposite angles)
  4. 4So triangle OAB is congruent to triangle ODC (SAS)

AnswerAO = DO (equal radii), BO = CO (equal radii), angle AOB = angle DOC (vertically opposite), so congruent (SAS)

Part (b) · 5 marks

With diameter 10 cm and AB = 9 cm, work out the gap between the circle's circumference and the perimeter of the shaded region.

  1. 1sin(1/2 × AOB) = 4.5/5, so angle AOB = 2 × sin⁻¹(0.9) = 128.31…°
  2. 2Angle AOD = 180 − 128.31… = 51.68…° = angle BOC
  3. 3Arcs DA + BC = 2 × 51.68…/360 × 10π = 9.02…
  4. 4Perimeter of shaded shape = 9 + 9 + 9.02… = 27.02…
  5. 5Difference = 10π − 27.02… = 4.395… ≈ 4.40

Answer4.39 or 4.40 or 4.392 to 4.398

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