Classwork Exercise and Series (Mathematics- SS3): Spherical Geometry And Bearing

Bearings

A directional compass is shown below.  It is used to find a direction or bearing.

The four main directions of a compass are known as cardinal points.  They are north (N), east (E), south (S) and west (W).  Sometimes, the half-cardinal points of north-east (NE), north-west (NW), south-east (SE) and south-west (SW) are shown on the compass.  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°

When using a directional compass, hold the compass so that the point marked north points directly away from you.  Note that the magnetic needle always points to the north.

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.

For example, 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).

Note:

The bearing of a point 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 with the point.

A bearing is used to represent the direction of one point relative to another point.

For example, the bearing of A from B is 065⁰.  The bearing of B from A is 245⁰.

Note:

Three figures are used to give bearings.

All bearings are measured in a horizontal plane.

Example

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°.

Direction

The conventional bearing of a point is stated as the number of degrees east or west of the north-south line.  We will refer to the conventional bearing simply as the direction.

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.

B from O is N60⁰W.

C from O is S70⁰E.

D from O is S80⁰W.

Note:

N30⁰E means the direction is 30⁰ east of north.

Example

Describe each of the following bearings as directions.
a.  076°
b.  150°
c.  225°
d.  290°

Solution:

a. The position of a point P on 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.

Questions

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

1. 

A. 140° B. 150⁰ C. 160⁰ D. 170⁰

2. 

A. 290° B. 300⁰ C. 320⁰ D. 340⁰

3. What is the bearing of C from B?

A.120⁰  B. 100⁰ C. 190⁰  D. 110⁰

4. What is the bearing of A from B?

A. 270⁰  B. 230⁰  C. 320⁰  D. 300⁰

5. A, B and C are three ships. The bearing of A from B is 045º. The bearing of C from A is 135º. If AB= 8km and AC= 6km, what is the bearing of B from C?

A. 236⁰  B. 245⁰  C. 262⁰  D. 260⁰

Answers

1. A  2. A  3. B  4. C  5. C

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