(a) Triangle PQR is right-angled at Q. PQ = 3a cm and QR = 4a cm. Determine PR in terms of a. (b) Ayo travels a distance of 24km from X on a bearing of 060°...

(a)

Since triangle \(PQR\) is right-angled at \(Q\), by Pythagoras' theorem,

(b)(i) Diagram

The diagram below is drawn to scale in the correct relative positions. \(XY=24\text{ km}\) is on a bearing of \(060^\circ\), \(YZ=18\text{ km}\), and \(XZ=30\text{ km}\).

(ii) Bearing of \(Z\) from \(Y\)

Therefore, \(\angle XYZ=90^\circ\). The bearing of \(X\) from \(Y\) is \(240^\circ\). Hence,

Bearing of \(Z\) from \(Y\) = \(150^\circ\).

(iii) Bearing of \(X\) from \(Z\)

Using the cosine rule at \(Z\),

Since the bearing of \(Y\) from \(Z\) is \(150^\circ+180^\circ=330^\circ\),

Bearing of \(X\) from \(Z\) = \(277^\circ\).