PK 2017 Q8
8. As shown in the diagram, \(FGHI\) is a trapezium with side \(GF\) parallel to \(HI\). The lengths of \(FG\) and \(HI\) are 50 and 20 respectively. The point \(J\) is on the side \(FG\) such that the segment \(IJ\) divides the trapezium into two parts of equal area.
What is the length of \(FJ\)?
- A 25
- B 30
- C 35
- D 40
- E 45
Pink Kangaroo 2017, Question 8. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: C 35
Let \(x\) be the length \(FJ\), and \(h\) be the height of the trapezium. Then the area of triangle \(FJI\) is \(\tfrac{1}{2}xh\) and the area of trapezium \(FGHI\) is \(\tfrac{1}{2}h(20 + 50) = 35h\). The triangle is half the area of the trapezium, so \(\tfrac{1}{2}xh = \tfrac{1}{2} \times 35h\), so \(x = 35\).
Solution © UK Mathematics Trust, from the PK 2017 solutions.