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0580/41 Oct/Nov 2021, Question 5: Circumference and Area of a Circle

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 11 marks

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The question, in short

Circle with centre O and points A, B, C, D on it; angle COD = 124° and angle BCO = 35°. Find angles CBD and BAD with reasons, then use a second circle with diameter QS, angle PRS = 42° and PQ = 5.9 cm to find the circumference.

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Part (ai) · 2 marks

Find the size of angle CBD, justifying the result with a stated geometrical reason.

  1. 1Angle CBD = 124 ÷ 2 = 62°
  2. 2Angle at the centre is twice the angle at the circumference

Answer62, angle at centre is twice angle at circumference

Part (aii) · 4 marks

Find angle BAD, giving a geometrical reason for every stage of the reasoning.

  1. 1OC = OD (radii), isosceles triangle: angle OCD = (180 − 124) ÷ 2 = 28°
  2. 2Angle BCD = 35 + 28 = 63°
  3. 3Angle BAD = 180 − 63 = 117° (opposite angles in a cyclic quadrilateral are supplementary)

Answer117, isosceles triangle, and opposite angles in a cyclic quadrilateral are supplementary

Part (b) · 5 marks

Using diameter QS, angle PRS = 42° and chord PQ = 5.9 cm, work out the circle's circumference.

  1. 1Angle PQS = angle PRS = 42° (angles in the same segment)
  2. 2Angle QPS = 90° (angle in a semicircle)
  3. 3QS = 5.9 ÷ cos 42° = 7.93… cm
  4. 4Circumference = π × 7.93… = 24.94… ≈ 24.9 cm

Answer24.9 or 24.94 to 24.95

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Also searched as: 0580/41/O/N/21 · 0580_w21_qp_41 · 0580 w21 41 · question 5.
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