7.5 Three Figure Bearing and its Applications

7.5 Three Figure Bearing and its Applications

A bearing is a way of describing the direction or angle from one point to another, measured in degrees from the north direction in a clockwise manner. It is always written as a three-digit angle between $000^\circ$ and $360^\circ$.

It is used in navigation, aviation, surveying and map reading to describe precise direction.

For example, a plane flying east has a bearing of $090^\circ$.

Example 12

In the diagram, the bearing of $Q$ from $P$ is $060^\circ$. Find the bearing of $P$ from $Q$.

[Diagram: points P and Q with bearing 060°]

Solution Extend the line $NQ$, so that angle $NQW = 180^\circ$

As alternate angles are equal, therefore

$m\angle PQW = 060^\circ$

So, the bearing of $P$ from $Q = 180^\circ + 060^\circ$

$= 240^\circ$

Example 13

Abdullah sails a ship from point $A$ to point $B$ on a bearing of $080^\circ$. What bearing should he follow to return from point $B$ back to point $A$?

Solution

The diagram shows the journey from $A$ to $B$.

[Diagram: journey from A to B on bearing 080°]

Extend the line of the journey and we get an angle of $80^\circ$ at $B$.

Bearing of $A$ from $B = 080^\circ + 180^\circ$

$= 260^\circ$

Example 14

An aeroplane departs from an airport and travels $40$ km due east, followed by $50$ km due north. The pilot then returns to the airport along the shortest possible route. What is the bearing and the distance of the shortest route?

[Diagram: east then north flight path with return bearing]

Solution

To find $\theta$

$\tan \theta = \dfrac{40}{50} = 0.8$

$\theta = \tan^{-1}(0.8)$

$= 38.66^\circ$

Bearing $= 180^\circ + 38.66^\circ = 218.66^\circ$

Suppose $x$ is the shortest route

$x^2 = 40^2 + 50^2$

$= 1600 + 2500$

$= 4100$

$x = \sqrt{4100}$

$= 64.03$ km

The pilot will fly on a bearing of $218.66^\circ$ for $64.03$ km to return to the airport.