Maclaurin 2018 Q4
4. The squares on each side of a right-angled scalene triangle are constructed and three further line segments drawn from the corners of the squares to create a hexagon, as shown. The squares on these three further line segments are then constructed (outside the hexagon).
The combined area of the two equal-sized squares is 2018 cm2.
What is the total area of the six squares?
Maclaurin Olympiad 2018, Question 4. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: 8072 cm2
Label the lengths as in the diagram below. Note that \(z = c\) since it is the hypotenuse of a right-angled triangle with sides \(a\) and \(b\). By chasing angles \(\alpha = \pi - A\) and \(\beta = \pi - B\). Hence \(\cos\alpha = -\dfrac{b}{c}\) and \(\cos\beta = -\dfrac{a}{c}\). We are told that \(2c^2 = 2018\) so \(c^2 = 1009\).
We have \(x^2 = b^2 + c^2 - 2bc\cos\alpha = 3b^2 + c^2\) by the cosine rule, and similarly \(y^2 = 3a^2 + c^2\).
Now the sum of the six squares is
By Pythagoras, \(a^2 + b^2 = c^2\) so the sum is \(8c^2\), which is 8072.
Note that the cosine rule can be avoided by the construction shown. This produces a right-angled triangle whose sides are \(a\) and \(2b\) whose hypotenuse is \(x\).
It follows that \(x^2 = a^2 + 4b^2\) and \(y^2 = b^2 + 4a^2\).
Solution © UK Mathematics Trust, from the Maclaurin Olympiad 2018 solutions.