Question Type 2: Exploring more complicated cases Exercises

From port $O$, Ship A sails $20\,\text{km}$ on bearing $065^\circ$ and Ship B sails $30\,\text{km}$ on bearing $120^\circ$. Find the bearing of A from B.

From $P$ to $Q$ is $10\,\text{km}$ on bearing $058^\circ$. From $Q$ to $R$ is $12\,\text{km}$ on bearing $128^\circ$. Find the distance and bearing of $R$ from $P$.

A plane flies $200\,\text{km}$ from $O$ on bearing $075^\circ$, then $160\,\text{km}$ on bearing $165^\circ$ to $B$. Find the distance $OB$ and the bearing of $B$ from $O$.

Points $A$ and $B$ are $15 \text{ km}$ apart with $B$ due north of $A$. From $A$, the bearing of a lighthouse $L$ is $030^\circ$; from $B$, the bearing of $L$ is $120^\circ$. Find the distance $AL$.

From point $P$, landmark $A$ is at distance $500\,\text{m}$ on bearing $042^\circ$, and landmark $B$ is at distance $650\,\text{m}$ on bearing $292^\circ$. Find the area of triangle $APB$.

From port $P$, Ship A travels at $14$ knots on bearing $045^\circ$ for $2$ hours. Ship B travels at $18$ knots on bearing $120^\circ$ for $1.5$ hours. Find the bearing of A from B.

Two coastal stations $A$ and $B$ are $8\,\text{km}$ apart on an east–west line, with $B$ due east of $A$. From $A$, the bearing of a ship $S$ is $040^\circ$; from $B$, the bearing of $S$ is $330^\circ$. Find the distance $AS$.

A ship sails $40\,\text{km}$ on bearing $030^\circ$, then turns $110^\circ$ to starboard and sails $25\,\text{km}$. Find the bearing of the ship from its starting point at the end of the second leg.

From point $P$, $Q$ is on a bearing of $300^\circ$ and $R$ is on a bearing of $040^\circ$. Find $\angle QPR$.

A ship sails $24\,\text{km}$ on a bearing of $040^\circ$, then $18\,\text{km}$ on a bearing of $310^\circ$. Find the distance and bearing of the ship from its starting point.

Two aircraft are observed from a radar station $O$. Aircraft $P$ is $50\,\text{km}$ from $O$ on bearing $220^\circ$; aircraft $Q$ is $70\,\text{km}$ from $O$ on bearing $310^\circ$. Find $\angle POQ$ and the distance $PQ$.

A drone flies from $O$ to $A$ on a bearing of $350^\circ$ for $12\,\text{km}$, then to $B$ on a bearing of $080^\circ$ for $9\,\text{km}$, then directly back to $O$.

Find the distance and bearing of the final leg (from $B$ to $O$).