IMC 2017 Q19
19. \(PQRS\) is a square. Point \(T\) lies on \(PQ\) so that \(PT : TQ = 1{:}2\). Point \(U\) lies on \(SR\) so that \(SU : UR = 1 : 2\). The perimeter of \(PTUS\) is 40 cm.
What is the area of \(PTUS\)?
- A 40 cm2
- B 45 cm2
- C 48 cm2
- D 60 cm2
- E 75 cm2
Intermediate Mathematical Challenge 2017, Question 19. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E 75 cm2
Let square \(PQRS\) have side \(3x\) cm. Then, as \(PT : TQ = 1 : 2\), \(PT = x\) cm. Similarly, \(US = x\) cm. In triangles \(PTS\) and \(UST\), \(PT = US\), \(\angle PTS = \angle UST\) (alternate angles) and \(TS\) is common to both. So the triangles are congruent (SAS).
Therefore \(UT = PS = 3x\) cm and \(\angle TUS = \angle SPT = 90\)°.
Hence \(PTUS\) is a rectangle, which has perimeter 40 cm. So \(40 = 2(3x + x) = 8x\). Therefore \(x = 5\) and the area of \(PTUS\), in cm2, is 15 × 5 = 75.
Solution © UK Mathematics Trust, from the IMC 2017 solutions. Longer versions with investigations: Solutions and investigations (PDF).