GK 2020 Q13
13. The pattern on a large square tile consists of eight congruent right-angled triangles and a small square. The area of the tile is 49 cm2 and the length of the hypotenuse \(PQ\) of one of the triangles is 5 cm. What is the area of the small square?
- A 1 cm2
- B 4 cm2
- C 9 cm2
- D 16 cm2
- E 25 cm2
Grey Kangaroo 2020, Question 13. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: A 1 cm2
Since the four rectangles are congruent, the diagonal \(PQ\) is also the side of a square. This square has area (5 × 5) cm2 = 25 cm2. Therefore the total area of the rectangles outside the square with side \(PQ\) but inside the large square is (49 − 25) cm2 = 24 cm2. However, this is also equal to the total area of the triangles inside the square with side \(PQ\). Therefore the area of the small square is (25 − 24) cm2 = 1 cm2.
Solution © UK Mathematics Trust, from the GK 2020 solutions.