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0580/22 Oct/Nov 2025, Question 11: Proving Triangles Are Similar

Cambridge IGCSE Mathematics 0580 · Paper 22 (Extended) · Non-calculator · 7 marks

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The question, in short

Using a diagram where S is on PQ, T is on PR and ST is parallel to QR, justify similarity of triangles PQR and PST, then use given lengths to find PQ and the area of QRTS in terms of k.

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

Part (a) · 3 marks

Give a geometric justification for each step showing that triangle PQR is similar to triangle PST.

  1. 1Angle PST = angle PQR, because they are corresponding angles (ST parallel to QR)
  2. 2Angle PTS = angle PRQ, because they are corresponding angles
  3. 3Angle SPT = angle QPR, because it is a common angle
  4. 4All three angles are equal, so the triangles are similar (AAA)

AnswerThree correct pairs with correct reasons: PQR = PST and corresponding angles; PRQ = PTS and corresponding angles; QPR = TPS and common angle. OR Two correct pairs with correct reasons and correct similarity condition e.g. AAA

Part (b) · 2 marks

With ST = 3 cm, QR = 9 cm and PS = 5 cm, calculate the length PQ.

  1. 1Scale factor = QR/ST = 9/3 = 3
  2. 2PQ = 3 × PS = 3 × 5 = 15 cm

Answer15

Part (c) · 2 marks

Given ST = 3 cm, QR = 9 cm and area of PST = 2k cm², express the area of QRTS in terms of k.

  1. 1Length scale factor = QR ÷ ST = 9 ÷ 3 = 3
  2. 2Area scale factor = 3² = 9
  3. 3Area PQR = 9 × 2k = 18k
  4. 4Area QRTS = 18k − 2k = 16k

Answer16k

Stuck on a step? Read the free lesson: Proving Triangles Are Similar

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Also searched as: 0580/22/O/N/25 · 0580_w25_qp_22 · 0580 w25 22 · question 11.
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