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0580/23 Oct/Nov 2024, Question 14: Angle at the Centre

Cambridge IGCSE Mathematics 0580 · Paper 23 (Extended) · Calculator · 4 marks

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The question, in short

Use circle theorems with a tangent at A and chords to find angles x and y, then find w using centre O and angle PQR = 115°.

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

Part (ai) · 1 mark

Find x, the angle ACB, using the tangent at A and angle SAD = 37°.

  1. 1Angles ACB and ADB stand on the same arc AB (angles in the same segment are equal)
  2. 2x = 41

Answer41

Part (aii) · 1 mark

Find y, the angle ABD, using angle ADB = 41° and the circle's properties.

  1. 1Alternate segment theorem: angle between tangent SA and chord AD = angle ABD
  2. 2y = 37

Answer37

Part (b) · 2 marks

Find w, the angle POR at centre O, given angle PQR = 115°.

  1. 1Opposite angles of a cyclic quadrilateral add to 180°: angle at the circle opposite Q = 180 − 115 = 65°
  2. 2Angle at the centre = 2 × angle at the circumference
  3. 3w = 2 × 65 = 130

Answer130

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Also searched as: 0580/23/O/N/24 · 0580_w24_qp_23 · 0580 w24 23 · question 14.
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