(a)
From the diagram and by Pythagoras theorem,
\(|PR|^{2} = |PQ|^{2} + |QR|^{2}\)
\(|PR|^{2} = (3a)^{2} + (4a)^{2}\)
\(|PR|^{2} = 9a^{2} + 16a^{2} = 25a^{2}\)
\(|PR| = \sqrt{25a^{2}} = 5a\)
(b)
(ii) From the figure, Bearing of Y from X = 060°
Using cosine rule,
\(\cos XYZ = \frac{24^{2} + 18^{2} - 30^{2}}{2 \times 24 \times 18}\)
\(\cos XYZ = 0 \implies < XYZ = 90°\)
\(< RYZ = 90° - 60° = 30°\)
\(< N_{2} YZ = 180° - 30° = 150°\) (angles on a straight line)
The bearing of Z from Y is 150°.
(iii) \(\cos XZY = \frac{30^{2} + 18^{2} - 24^{2}}{2 \times 30 \times 18}\)
= \(\frac{900 + 324 - 576}{1080}\)
\(\cos XZY = \frac{648}{1080} = 0.6\)
\(< XZY = \cos^{-1} (0.6) = 53.13°\)
\(\therefore\) The bearing of X from Y = 360° - (53.13° + 30°) = 276.87°.