IMC 2017 Q13
13. The diagram shows a quadrilateral \(PQRS\) made from two similar right-angled triangles, \(PQR\) and \(PRS\). The length of \(PQ\) is 3, the length of \(QR\) is 4 and \(\angle PRQ = \angle PSR\).
What is the perimeter of \(PQRS\)?
- A 22
- B \(22\frac{5}{6}\)
- C 27
- D 32
- E \(45\frac{1}{3}\)
Intermediate Mathematical Challenge 2017, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 22
Pythagoras' Theorem shows that \(PR = \sqrt{3^2 + 4^2} = 5\). So the perimeter of triangle \(PQR\) is 12. Since the triangles are similar and \(PR : PQ = 5 : 3\) we see that the perimeter of triangle \(PRS\) is 20. Hence the perimeter of \(PQRS\) is \(12 + 20 - 2 \times PR = 32 - 10 = 22\).
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).