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0580/23 May/Jun 2021, Question 19: Cyclic Quadrilaterals

Cambridge IGCSE Mathematics 0580 · Paper 23 (Extended) · Calculator · 5 marks

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The question, in short

Circle with tangent ATB at T, chords PT, PQ, QT and PT = PQ; angle ATP = 50°. Find angle TPQ, then use a cyclic quadrilateral with exterior angle 68° and angles 3x°, 2x°, w° to find w and x.

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Part (a) · 2 marks

Using the tangent-chord angle 50° and equal chords PT and PQ, find angle TPQ.

  1. 1Angle PQT = angle ATP = 50° (alternate segment theorem)
  2. 2PT = PQ so triangle PTQ is isosceles: angle PTQ = angle PQT = 50°
  3. 3Angle TPQ = 180 − 50 − 50 = 80°

Answer80

Part (b) · 3 marks

Cyclic quadrilateral with exterior angle 68° and interior angles 3x°, 2x°, w°; find w and x.

  1. 1Interior angle at top right = 180 − 68 = 112°
  2. 2w is opposite 112°, so w = 180 − 112 = 68
  3. 33x and 2x are opposite angles: 3x + 2x = 180
  4. 45x = 180, x = 36

Answerw = 68, x = 36

Stuck on a step? Read the free lesson: Cyclic Quadrilaterals

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Also searched as: 0580/23/M/J/21 · 0580_s21_qp_23 · 0580 s21 23 · question 19.
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