In the diagram,

Explanation

(a) 

Let (< SQR = theta ; < QRS = 90° – 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°)

(therefore < QRS = 90° – 53.1° = 36.9°)

(approxeq 37°) (to the nearest degree)

(ii) In (Delta) PQS,

(|PQ|^{2} = 12^{2} + 9^{2})

= (144 + 81 = 225)

(|PQ| = sqrt{225} = 15 cm)