Bearings

Further Ideas

A particularly important application of bearings is fixing a position by triangulation. This involves taking a bearing from, or to, each of two known positions in order to establish a third unknown position in relation to them. The three positions can be considered as vertices of a triangle. Knowing two angles is sufficient to fix the third. Provided that at least one length is known, the triangle can be solved to find the unknown lengths. This can be done using scale drawing, measurement on a map or trigonometry. This technique is widely used in navigation, surveying and tracking systems.

The following examples could be used in the classroom to demonstrate how bearings can be used to fix position. Students will need to use back bearings or trial and improvement to solve these.

1. Draw coordinate axes, with both the x and y axes marked from 0 to 10. Mark the direction of north parallel to the y-axis. Plot and label the points A (1,8), B (4,4) and C (10,10).

Suppose that, measured from a certain ship:

• The bearing of A is 287º.
• The bearing of B is 238º.
• The bearing of C is 037º.

Find the coordinates of the position of the ship.

How many of the points A, B and C do you actually need to find the ship? Why do you think three are given in the question?

Place the ship somewhere else on the grid and give a partner sufficient information to locate it.

2. Use a scale drawing to solve the following problem.

The captain of a ship in distress radios the coastguard with the information that he can see a lighthouse on the coast, on a bearing of 140º. The coastguard knows that the lighthouse is 15km due east of his post. The coastguard asks the captain to set off a distress flare and when he sees it he estimates the bearing of the ship from his position to be 36º. The coastguard can now find how far the ship is from his position and organise assistance. Find this distance.

What if the ship’s radio wasn’t working, but both the coastguard and the lighthouse keeper spotted the distress flare? On what bearing would the ship be seen from the lighthouse? Can the coastguard still locate the ship accurately?

What if the lighthouse is not due east, but on a bearing of 100º from the coastguard’s station?

A simple practical exercise could help students to appreciate the technique. Two students stand a known distance apart (for example, using markings on the school field or playground), and each uses a compass to find the bearing of a third student standing some distance away. They can then draw the triangle and measure the distance of their partner from their position. If necessary the task can be simplified by placing the first two students facing north, a fixed or measurable distance apart, on a line running east-west.

Alternatively, if local maps are available, students could take bearings of two visible landmarks and use the information to mark their own position on the map.