## Bearings: Three Key Rules

**Bearings** are a way of expressing the angle between two objects. There is a specific set of rules about how bearings should be calculated and expressed.

1. Always measure bearings from the **North line**.

2. Always express your answers as **three-figure bearings** (so 60\degree would be 060\degree).

3. Always draw and measure bearings **clockwise**.

Understanding angles on parallel lines is required for this topic.

## Example 1: Measure a Bearing

Find the bearing of \textcolor{blue}{B} from \textcolor{red}{A}.

**[1 mark]**

**Note:** the terminology “B from A” is always used, as opposed to “A to B”.

So, we have our two points and a North line coming off both.

The bearing of \textcolor{blue}{B} from \textcolor{red}{A} is measured from the North line going clockwise until we hit the straight line.

Then using a protractor, we measure the angle to be 110\degree which is the bearing of \textcolor{blue}{B} from \textcolor{red}{A}.

**Example 2: Drawing Bearings**

Two boats \textcolor{red}{A} and \textcolor{blue}{B} are 5km apart, and the bearing of \textcolor{blue}{B} from \textcolor{red}{A} is 256\degree.

Using the scale 1\text{ cm}:1\text{ km}, construct a diagram showing the relative positions of points \textcolor{red}{A} and \textcolor{blue}{B}.

**[2 marks]**

First, we draw point \textcolor{red}{A} with a North line and measure an angle of 104\degree going anticlockwise from it (This is because 360 - 254 = 104\degree. You can’t measure 256\degree using a protractor any other way).

Then, as \textcolor{red}{A} and \textcolor{blue}{B} are 5 km apart, we will need to make the line from \textcolor{red}{A} to \textcolor{blue}{B} (going along the bearing we’ve determined) 5 cm long.

The result of this is below, not drawn accurately.

**Example 3: Finding Bearings**

The diagram below shows the bearing of \textcolor{blue}{B} from \textcolor{red}{A}.

Find the bearing of \textcolor{red}{A} from \textcolor{blue}{B}.

**[2 marks]**

Now, we can’t measure the angle because the diagram is not drawn accurately.

We will use the fact that both North lines are parallel and extend the line \textcolor{red}{A}\textcolor{blue}{B} past point \textcolor{blue}{B}, the angle formed by the North line at \textcolor{blue}{B} and the extension to line \textcolor{red}{A}\textcolor{blue}{B} and the bearing of \textcolor{blue}{B} from \textcolor{red}{A} are **corresponding** angles (also known as an “F angle”).

So, from our knowledge of **parallel lines**, we know that they must be equal.

Finally, we are measuring the line of \textcolor{red}{A} from \textcolor{blue}{B} so we need to go clockwise from the north line at \textcolor{blue}{B} to the line \textcolor{red}{A}\textcolor{blue}{B}.

We have 94\degree but need the remaining portion of the angle.

Fortunately, the remaining portion of the angle is just a straight line, so the bearing of \textcolor{red}{A} from \textcolor{blue}{B} is

94 + 180 = 274\degree

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### Example Questions

**Question 1:** A boat and a lighthouse are 70 miles apart. The bearing of the lighthouse from the boat is 051\degree.

Using the scale 1 cm : 10 miles, construct a diagram showing the relative positions of the lighthouse and the boat.

**[2 marks]**

Let the lighthouse be L and the boat be B. As we’re finding the bearing of L from B, we shall measure an angle of 051\degree clockwise at B.

Then, as B and L are 70 miles apart, we will need to make the line from B to L 7cm long. The final diagram should look like,

**Question 2:** The diagram below shows the bearing of A from B. Find the bearing of B from A.

**[2 marks]**

We can find the other angle around the point B by subtracting 295 from 360,

360\degree - 295\degree = 65 \degree

Then, because the two North lines are parallel, we can say that the bearing of B from A and the 65\degree angle we just found are **co-interior. **These two angles (marked with red below) must add to 180.

So, we get:

\text{Bearing of B from A } = 180\degree - 65\degree = 115\degree

**Question 3: **The location C is on a bearing of 140 \degree from A. The bearing of C from B is 250 \degree. Find the location C and mark it on the diagram below.

**[2 marks]**

Drawing straight lines along each of the bearings, we can find C at the point of intersection of both lines.

**Question 4:** By measuring the diagram given, state the bearing of B from A.

**[1 mark]**

By use of a protractor or otherwise we find the angle of 60 degrees. This written as a bearing is,

060 \degree

**Question 5: **A diagram of a bearing is shown below. Given that the bearing of B from A is 060 \degree, state the bearing of A from B.

**[3 marks]**

The two North lines are parallel, so we can say that the bearing of B from A and the **co-interior **angle at B must add to 180 degrees. Thus the co-interior angle is

180 \degree-60\degree = 120 \degree

As angles around a point sum to 360 degrees we can find the bearing of A from B as,

360 \degree-120 \degree=240 \degree

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