(a) Let \(< SQR = \theta ; < QRS = 90^\circ - \theta ; < QPS = \theta\) In \(\Delta\) PQS, \(\tan \theta = \frac{x}{9} ... (1)\) In \(\Delta\) QRS, \(\tan \theta = \frac{16}{x} ... (2)\) From (1) and (2), \(\frac{x}{9} = \frac{16}{x}\) \(x^{2} = 16 \times 9 \implies x = \sqrt{16 \times 9} = 12 cm\) (b) (i) From \(\Delta\) QRS, \(\tan \theta = \frac{16}{12} = 1.333\) \(\theta = \tan^{-1} (1.333) = 53.1^\circ\) \(\therefore < QRS = 90^\circ - 53.1^\circ = 36.9^\circ\) \(\approxeq 37^\circ\) (to the nearest degree) (ii) In \(\Delta\) PQS, \(|PQ|^{2} = 12^{2} + 9^{2}\) = \(144 + 81 = 225\) \(|PQ| = \sqrt{225} = 15 cm\)