Geometry

NZ Level 5

Bearings

Lesson

In surveying and air navigation, bearing are used to help identify the location of an object.

The four main directions of a compass are known as cardinal directions. They are north (N), east (E), south (S) and west (W).

A three-figure bearing are:

- measured from north ($N$
`N`) - measured in a clockwise direction
- written using three figures

A $T$`T` is often but not always used to indicate a true bearing. If the angle measure is less than $100^\circ$100° it would be written something like 040° or 040°T.

To use true bearing to describe the location of a plane at point $B$`B` from the airport at point $A$`A` :

- place the centre of a compass on the starting point, in the case the airport.
- starting at North, rotate clockwise until we get to the line $AB$
`A``B`. - write angle as the true bearing of point $B$
`B`.

The true bearing of $B$`B` from $A$`A` is $127^\circ$127° or $127^\circ$127°$T$`T`.

The diagram below describes the bearing of $P$`P` from $O$`O`. Rotating clockwise from North, we get an angle of $55^\circ$55°.

Since this measure is less than three digits, we put a 0 in front of it so the true bearing of $P$`P` is 055°. Consider the true bearing of $O$`O` from $P$`P` . Since angle of elevation is equal to angle of depression and we are starting at $P$`P` the true bearing would be $180+55=235^\circ$180+55=235°.

A compass bearing describes the location of a point using:

- the starting direction of either north or south;
- the acute angle needed to rotate
- the direction to rotate, east or west.

To describe the position of point $B$`B` from $A$`A` we would say:

"Starting at South, I then rotate $53$53° towards East."

We can write this mathematically as:

$S$`S`$53$53°$E$`E`

Find the **three-figure** and the **compass bearings** of point $P$`P` from $O$`O`.

Solution:

Starting at North rotate in a clockwise direction.

$360^\circ-47^\circ=313^\circ$360°−47°=313°

The three-figure bearing of $P$`P` from $O$`O` is $313^\circ$313°$T$`T`.

Point $P$`P` is closest to North, so starting at North, rotate $47^\circ$47° towards West.

The compass bearing of $P$`P` from $O$`O` is $N$`N`$47^\circ$47°$W$`W`.

The bearing needed or used completely depends on which position comes first. Have a look at the investigation below, it quickly shows you how the angle changes depending on if we are measuring the bearing of A from B or B from A.

Let's have a look at these worked examples.

Consider the point $A$`A`.

Find the true bearing of $A$

`A`from $O$`O`.What is the compass bearing of point $A$

`A`from $O$`O`?$\editable{}$ $\editable{}$$^\circ$° $\editable{}$

What is the true bearing of Southwest?

In the figure below, point $B$`B` is due East of point $A$`A`. We want to find the position of point $A$`A` relative to point $C$`C`.

Find the true bearing of point $A$

`A`from point $C$`C`.What is the compass bearing of point $A$

`A`from point $C$`C`?$\editable{}$ $\editable{}$$^\circ$° $\editable{}$

Interpret points and lines on co-ordinate planes, including scales and bearings on maps