# Form 4 & 5 Unit 10 Lesson 16: Introduction to Bearings

### Objectives

At the end of the lesson, students should be able to;

• Demonstrate a clear understanding of bearings.
• Calculate the magnitude and the direction of any point.

There are many situations in which you might need to describe your position and direction of travel. For example, you may tell a friend that you will meet them by the bus station, at 7pm, and that if they are late you will start walking up the high street. In mathematics, we use more precise ways to describe position and direction of travel.

Angles have many applications in our everyday lives. Builder architects for example use angles in their work. In geography angles are used to show directions. Sailors have no road to follow just a compass and a map to navigate with.

Surveyors also use angles and bearing in their work. Bearing gives direction in terms of an angle. The above compass shows degree measurements from 0° to 360° in 10° intervals with:

• North representing 0° or 360°
• East representing 90°
• South representing 180°
• West representing 270°

The directions half way between the cardinal points are marked in the figure below. We state North (N) or South(S) first and then the angle we turn through towards the East (E) or the West (W)

To state the direction of a point, write:

• N or S which is determined by the angle being measured.
• The angle between the North or South line and the point, measured in degrees.
• E or W which is determined by the location of the point relative to the North-South line E.g. In the above diagram, the direction of:

• A from O is N300E.
• from O is N600W.
• from O is S700E.
• from O is S800W.

Example 1

Represent N500E and S400W on different cardinals.

Solution

N500E means from N measure 500 towards E.    S400W means from S measure 400 towards W. True Bearing

The true bearing to a point is the angle measured in degrees in a clockwise direction from the north line.  We will refer to the true bearing simply as the bearing.

A three figure bearing is always;

• An angle measured in degrees.
• Measured as angles in the clockwise direction from the geographic North.
• An angle given in three figure or digit from 0000 to 3600.

Note

You use extra zeros to make the number up to three digits if you need to.

For example, 30 gives a bearing of 0030and 990 gives a bearing of 0990.

Example 2
Look at the diagram below:

State the bearing of P and the bearing of Q. Solution

The bearing of point P is 0650 which is the number of degrees in the angle measured in a clockwise direction from the North line to the line joining the centre of the compass at O with the point P (i.e. OP).

The bearing of point Q is 3000 which is the number of degrees in the angle measured in a clockwise direction from the north line to the line joining the centre of the compass at O with the point Q (i.e. OQ).

Example 3

State the bearing of the point P in each of the following diagrams: Solution

a) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P.

The bearing of point P is 048°. b) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P.

The cardinal point S corresponds to 180°.  It is clear from the diagram that the required angle is 60° larger than 180°.  So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 180° + 60° = 240°.

So, the bearing of point P is 240°. c) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P.

The cardinal point S corresponds to 180°.  It is clear from the diagram that the required angle is 40° less than 180°.  So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 180°40° = 140°.

So, the bearing of point P is 140°. d) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point P.

The cardinal point W corresponds to 270°.  It is clear from the diagram that the required angle is 20° larger than 270°.  So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point P is 270° + 20° = 290°.

So, the bearing of point P is 290°. Example 4
Points of the compass can all be converted into bearings.

Find the bearings for:
(a) East (E)
(b) South (S)
(c) South-East (SE)

Solution REMEMBER: Bearings are always measured in a clockwise direction from the North and are given as 3 digits.

Example 5

Describe each of the following bearings as directions.

a) 076°

b) 150°

c) 225°

d) 290°

Solution

a) The position of a point Pon a bearing of 076° is shown in the following diagram. The position of the point P is 76° east of north.  So, the direction is N76°E.

b) The position of a point P on a bearing of 150° is shown in the following diagram The position of the point P is 180°  150° = 30° east of south.  So, the direction is S30°E.

c) The position of a point P on a bearing of 225° is shown in the following diagram. The position of the point P is 225°  180° = 45° west of south.  So, the direction is S45°W.

d) The position of a point P on a bearing of 290° is shown in the following diagram. The position of the point P is 360°  290° = 70° west of north.  So, the direction is N70°W.

Example 6

Give:

(i) the compass bearing and the

(ii) true bearing of A from O for each of the following. Solution

a) (i) compass bearing is N800W

(ii)True bearing of A from O = 3600– 800

= 2800

b) (i) compass bearing =1800– 1600

= 200

Compass bearing = S200E

(ii) True bearing of A from O = 1600T

c) (i) OA is half-way between south and west.

Compass bearing is S450W

1800+ 450 = 2250

(ii) True bearing of A from O = 2250

Unit 10 Lesson 16: Exercise
What angle do you turn through if you turn clockwise from:

1. N to S
2. E to W
3. N to NE
4. S to NE
5. W to NW

6) For each of the following match the compass bearing of A from O below with the diagram supplied

N800W       S150W       NW       due north      SE         S300W 7)  Match each of the diagrams in question (6) to the corresponding true bearing.

1. a) 2800 T   b) 3150          T c) 1350 T          d) 0000T                  e) 2100 T       f) 1950T

8) Give

(i) the compass bearing

(ii) true bearing of following 9) Draw a diagram to show each of the following compass bearings:

a) S300E b) S400W   c) N300E      d) N560W     e) N200E       f) S670E

10) Draw a diagram to show each of the following true bearings.

a) 3000T b) 3400T     c) 3040T       d) 1700T      e) 1450T     f) 0140T

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