Measurement, STD1 2026 HSC 31

A ship sails 8 km from \(A\) to \(B\) on a bearing of 070°. It then sails from \(B\) to \(C\).

Point \(D\) lies on the line \(CE\) and is 5 km due east of \(B\), and \(E\) is 8.7 km due south of \(C\) and directly east of \(A\).

What is the true bearing of \(A\) from \(C\), correct to the nearest degree? (4 marks)

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\(235^{\circ}\)

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\(\text{In}\ \Delta ABF:\)

\(\angle BAF=90-70=20^{\circ}\)

\(\cos 20^{\circ}\)	\(=\dfrac{AF}{8}\)
\(AF\)	\(=8\cos 20^{\circ}=7.517…\ \text{km}\)

\(FE=BD=5\ \text{km}\)

\(AE=7.517…+5=12.517…\ \text{km}\)

\(\text{In}\ \Delta ACE:\)

\(\tan \angle ACE\)	\(=\dfrac{12.517…}{8.7}\)
\(\angle ACE\)	\(=55.19…^{\circ}\)

\(\text{Bearing of}\ A\ \text{from}\ C\)	\(=180+55.19…\)
	\(=235^{\circ}\ \text{(nearest degree)}\)