2025 AHS PRELIMS P2 Q5

In triangle \(ABC\), angle \(CDE\) and angle \(CEB\) are right angles. \(DE=2.5\text{ cm}\), \(CD=5\text{ cm}\), \(AE=x\text{ cm}\) and \(CE=y\text{ cm}\).

1. 1. Show that \(\triangle ECD\) and \(\triangle ACE\) are similar.[1]
   2. Express \(y\) in terms of \(x\).[1]

2. 1. In triangle \(PQR\), \(\angle PQR=30^\circ\), \(PQ=x+3\text{ cm}\) and \(QR=y+2\text{ cm}\). Express the area of triangle \(PQR\) in terms of \(x\) and \(y\).[1]
   2. The area of triangle \(PQR\) is \(17\dfrac12\text{ cm}^2\). Using your answers in (a)(ii) and (b)(i), find the value of \(x\) and of \(y\).[3]