GK 2019 Q24
24. The diagram shows the square \(WXYZ\). The points \(P\), \(Q\) and \(R\) are the midpoints of the sides \(ZW\), \(XY\) and \(YZ\) respectively. What fraction of the square \(WXYZ\) is shaded?
- A \(\dfrac{3}{4}\)
- B \(\dfrac{5}{8}\)
- C \(\dfrac{1}{2}\)
- D \(\dfrac{7}{16}\)
- E \(\dfrac{3}{8}\)
Grey Kangaroo 2019, Question 24. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: E \(\dfrac{3}{8}\)
Label the intersection of \(WQ\) and \(XP\) as \(V\) and the midpoint of \(WX\) as \(U\). Let the side-length of the square be 1 unit. The area of triangle \(WXR\) is \(\tfrac{1}{2} \times 1 \times 1\) units2 = \(\tfrac{1}{2}\) units2. Consider triangle \(WXP\) and triangle \(UXV\). These two triangles have the same angles and hence are similar. Since \(UX\) is half of \(WX\), it follows that \(VU\) is half of \(PW\) and hence has length \(\tfrac{1}{4}\) unit. Therefore the area of triangle \(WXV\) is \(\left(\tfrac{1}{2} \times 1 \times \tfrac{1}{4}\right)\) units2 = \(\tfrac{1}{8}\) units2. Hence the shaded area is \(\left(\tfrac{1}{2} - \tfrac{1}{8}\right)\) units2 = \(\tfrac{3}{8}\) units2. Therefore the fraction of the square that is shaded is \(\tfrac{3}{8}\).
Solution © UK Mathematics Trust, from the GK 2019 solutions.