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4024/22 Oct/Nov 2022, Question 10: Combining Circle Theorems

Cambridge O Level Mathematics 4024 · Paper 22 (Calculator) · Calculator · 10 marks

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The question, in short

Circle geometry: prove triangles ABC and CDA congruent, explain why ABCD is a rectangle, then calculate a shaded area using given angles and radius 6 cm.

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

Part (ai) · 3 marks

Show triangle ABC congruent to triangle CDA, justifying each statement with a reason.

  1. 1AC is common to both triangles
  2. 2Angle CAB = angle ACD = 40° (given)
  3. 3Angle ABC = angle CDA = 90° (angle in a semicircle, AC is a diameter)
  4. 4So triangle ABC is congruent to triangle CDA (AAS)

AnswerAC is common; angle ACD = angle CAB (given); angle ABC = angle CDA (= 90°) angle in semicircle; congruent (AAS)

Part (aii) · 2 marks

Explain why quadrilateral ABCD must be a rectangle.

  1. 1Angle ACD = angle CAB are alternate angles, so AB is parallel to DC
  2. 2Angle ABC = angle ADC = 90° (angle in a semicircle)
  3. 3So ABCD has parallel sides at right angles: it is a rectangle

AnswerOption 1: angle ACB = angle CAD = 50° since angle ABC = angle CDA = 90° angle in semicircle; rectangle because angle DAB = angle BCD = angle ABC = angle CDA = 90° Option 2: AB parallel to DC since angle ACD and angle CAB are alternate angles; angle ABC = angle ADC = 90° angle in semicircle; rectangle because perpendicular parallel sides

Part (b) · 5 marks

Calculate the shaded area between chord HG and minor arc HG, using angle EHO = 30°, angle EFG = 116° and radius 6 cm.

  1. 1Angle EHG = 180 − 116 = 64° (opposite angles of a cyclic quadrilateral)
  2. 2Angle OHG = 64 − 30 = 34°
  3. 3OH = OG, so angle HOG = 180 − 2 × 34 = 112°
  4. 4Sector = 112/360 × π × 6² = 35.18…; triangle = 1/2 × 6² × sin 112° = 16.68…
  5. 5Shaded area = 35.18… − 16.68… = 18.49… ≈ 18.5 cm²

Answer18.5 or 18.49 to 18.50…

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Also searched as: 4024/22/O/N/22 · 4024_w22_qp_22 · 4024 w22 22 · question 10.
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