PK 2015 Q11
11. The square \(FGHI\) has area 80. Points \(J\), \(K\), \(L\), \(M\) are marked on the sides of the square so that \(FK = GL = HM = IJ\) and \(FK = 3KG\).
What is the area of the shaded region?
- A 40
- B 35
- C 30
- D 25
- E 20
Pink Kangaroo 2015, Question 11. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 25
Let \(l\) be the length of \(KG\). Then \(FK = 3l\), and the sides of the square \(FGHI\) are each \(4l\).
Since the area of the square is 80, we have \((4l)^2 = 80\), which is \(16l^2 = 80\); hence \(l^2 = 5\).
By Pythagoras' Theorem, \(JK^2 = FJ^2 + FK^2 = l^2 + (3l)^2 = 10l^2\).
The shaded area is half the area of the square \(JKLM\), i.e. half of \(JK^2 = \tfrac{1}{2} \times 10l^2 = 5l^2 = 5 \times 5 = 25\).
Solution © UK Mathematics Trust, from the PK 2015 solutions.