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0580/42 May/Jun 2023, Question 1: Cyclic Quadrilaterals

Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 18 marks

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The question, in short

Use the given diagrams and angle facts to find unknown angles, a ratio value, a fraction of angles, prove a quadrilateral is cyclic, and calculate a major arc length.

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

Part (a) · 3 marks

Isosceles triangle with apex 42° and base extended; find exterior angle x.

  1. 1Base angles = (180 − 42)/2 = 69°
  2. 2x = 180 − 69
  3. 3x = 111

Answer111

Part (b) · 3 marks

Three angles a, b, c in ratio 3:4:5 around a point; find c.

  1. 1Angles at a point add to 360°
  2. 23 + 4 + 5 = 12 parts
  3. 3c = 5/12 × 360
  4. 4c = 150

Answer150

Part (c) · 4 marks

Regular pentagon exterior angle d and regular hexagon interior angle h; find d/h in simplest form.

  1. 1d = 360 ÷ 5 = 72°
  2. 2h = 180 − 360 ÷ 6 = 120°
  3. 3d/h = 72/120
  4. 4= 3/5

Answer3/5

Part (d) · 5 marks

Quadrilateral PQRS with angles x, x+20, 3x−40, 2x−5; show it is cyclic.

  1. 1x + (2x − 5) + (x + 20) + (3x − 40) = 360
  2. 27x − 25 = 360, so 7x = 385
  3. 3x = 55
  4. 4Angle S + angle Q = 55 + 125 = 180
  5. 5Angle P + angle R = 105 + 75 = 180
  6. 6Opposite angles sum to 180°, so PQRS is a cyclic quadrilateral

Answerx = 55; angles 55°, 105°, 125°, 75°; opposite angles sum to 180°, so PQRS is cyclic

Part (e) · 3 marks

Circle radius 9 cm, minor sector angle 50° removed; calculate length of major arc AB.

  1. 1Major sector angle = 360 − 50 = 310°
  2. 2Arc = 310/360 × 2 × π × 9
  3. 3= 48.69…
  4. 448.7 cm

Answer48.7 or 48.69 to 48.70…

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Also searched as: 0580/42/M/J/23 · 0580_s23_qp_42 · 0580 s23 42 · question 1.
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