PK 2015 Q20
20. In the triangle \(FGH\), we can draw a line parallel to its base \(FG\), through point \(X\) or \(Y\). The areas of the shaded regions are the same. The ratio \(HX : XF = 4 : 1\). What is the ratio \(HY : YF\) ?
- A 1 : 1
- B 2 : 1
- C 3 : 1
- D 3 : 2
- E 4 : 3
Pink Kangaroo 2015, Question 20. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 3 : 2
In the triangle on the left, the unshaded triangle is similar to triangle \(FGH\), and is obtained from it by a scale factor of \(\tfrac{4}{5}\). Hence its area is \(\left(\tfrac{4}{5}\right)^2 = \tfrac{16}{25}\) of the area of \(FGH\). The shaded area is therefore \(\tfrac{9}{25}\) of the area of \(FGH\). Hence \(HY : HF = 3 : 5\) and so \(HY : YF = 3 : 2\).
Solution © UK Mathematics Trust, from the PK 2015 solutions.