Bearings Constructions And Scale Drawings

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Notes

Bearings

Bearings describe an angle used in navigation, measured from **North** (usually straight up on a diagram) **clockwise**, and written with **three digits** (e.g., 059°).
To find the bearing of A from B: start at B, draw a North line, then measure the clockwise angle to the line joining B to A.
To find the bearing of B from A: start at A, draw a North line, then measure the clockwise angle to the line joining A to B.
If the bearing of A from B is less than 180°, add 180° to get the bearing of B from A; if it is more than 180°, subtract 180°.
Always draw a clear, large diagram and label angles carefully; use a protractor for accurate angle measurement.

Scale

A **scale** is a ratio comparing drawn size to real-life size; e.g., 1:50 000 means 1 cm on the drawing represents 50 000 cm in reality.
To find actual distance from a map: measure the drawn length, multiply by the scale factor, then convert units (e.g., cm → m → km).
To find drawn length from actual distance: convert actual distance to the same units as the scale, then divide by the scale factor.
Common sense checks: when converting to smaller units (e.g., km → m), the number gets larger; when converting to larger units, it gets smaller.

Constructing SSS Triangles

**SSS triangle construction** uses the lengths of all three sides; you need a pencil, ruler, and compasses.
Step 1: Draw the **longest side** as a horizontal base near the bottom of your space, label its length.
Step 2: Set compasses to the length of one remaining side, place the point at one end of the base, and draw an **arc** above the base.
Step 3: Without changing compass width, set to the third side, place the point at the other end of the base, and draw another arc crossing the first.
Step 4: Use a ruler to draw straight lines from each end of the base to the intersection point of the arcs.
Step 5: Check the two new sides match the given lengths; **do not erase construction arcs** – they are required for marks.

Drawing a Point on a Bearing

To plot a point B from A on a given bearing and distance: draw a North line at A, measure the bearing clockwise from North, draw a line in that direction.
Use the scale to convert the real distance to the drawing length, then measure along the bearing line to mark point B.
Always label North lines and points clearly; use a sharp pencil for accuracy.

Back Bearings

A **back bearing** is the bearing in the opposite direction; it differs by 180°.
If the original bearing is less than 180°, add 180° to get the back bearing; if it is more than 180°, subtract 180°.
Example: bearing of B from A is 105°, so bearing of A from B is $105^{\circ} + 180^{\circ} = 285^{\circ}$ .

Using Bearings with Trigonometry

Trickier bearing problems may require **Pythagoras** or **trigonometry** to find missing distances or angles.
Always draw a diagram if not given; annotate all known angles and lengths.
Remember that bearings are measured from North, so angles in triangles may need to be calculated using alternate angles or co-interior angles.

Bearing Measurement

SSS Triangle Construction

Scale Conversion

Back Bearing

Practice questions

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1.The bearing of a ship from a lighthouse is 050°. Work out the bearing of the lighthouse from the ship.
Easy
- A050°
- B130°
- C230°
- D310°
2.The scale of a map is 1 cm to 3 km. On the map, the distance between two towns is 4.5 cm. Find the actual distance.
Easy
- A1.5 km
- B7.5 km
- C13.5 km
- D15 km
3.Which of the following is a correct rule for bearings?
Easy
- ABearings are measured anticlockwise from North.
- BBearings are measured clockwise from South.
- CBearings are measured clockwise from North and written as three-digit numbers.
- DBearings are measured anticlockwise from South and written as three-digit numbers.
4.Martin runs from checkpoint A to checkpoint B on a bearing of 065°. What is the bearing of A from B?
Easy
- A065°
- B115°
- C245°
- D295°
5.The bearing of B from A is 105°. Find the bearing of A from B.
Medium
- A075°
- B105°
- C195°
- D285°
6.A map has scale 1 : 25000. A lake has area $33.6 cm^{2}$ on the map. Calculate the actual area of the lake in $km^{2}$ .
Medium
- A $0.021 km^{2}$
- B $0.21 km^{2}$
- C $2.1 km^{2}$
- D $21 km^{2}$
7.In triangle $PQR, QR = 10$ cm and $PR = 11$ cm. Using a ruler and compasses only, which of the following is the correct first step to construct triangle PQR if PQ is already drawn?
Medium
- ADraw a circle of radius 10 cm around P.
- BDraw a circle of radius 11 cm around Q.
- CDraw a circle of radius 10 cm around Q and a circle of radius 11 cm around P.
- DDraw a circle of radius 11 cm around P and a circle of radius 10 cm around Q.
8.The bearing of D from A is 070°. Angle $CAD = 35.8^{\circ}$ . Find the bearing of A from C.
Medium
- A034.2°
- B124.2°
- C214.2°
- D304.2°

Summary

## Bearings, Constructions & Scale Drawings ### Bearings - **Rules:** - Measured from **North** (label North line). - Measured **clockwise**. - Written as **3 digits** (e.g., 059°). - **Finding a bearing:** - "Bearing of A from B" → start at B, draw North line, measure clockwise to A. - "Bearing of B from A" → start at A, draw North line, measure clockwise to B. - **Reverse bearing:** - If bearing of A from B < 180° → add 180° to get bearing of B from A. - If > 180° → subtract 180°. - **Drawing a point on a bearing:** 1. Draw North line at starting point. 2. Measure angle clockwise from North. 3. Draw line and mark point at given distance (use scale). ### Scale - **Scale ratio** (e.g., 1 : 50 000) means 1 unit on drawing = 50 000 units in real life. - **Finding actual distance:** 1. Measure distance on drawing (cm). 2. Multiply by scale factor. 3. Convert units (e.g., cm → m → km). - **Finding drawing distance:** 1. Convert actual distance to same units as scale. 2. Divide by scale factor. - **Common conversions:** 1 km = 1000 m = 100 000 cm. ### Constructing Triangles (SSS) - **Given three side lengths:** 1. Draw longest side as base using ruler. 2. Set compass to one side length, draw arc from one endpoint. 3. Set compass to third side length, draw arc from other endpoint to intersect first arc. 4. Draw lines from endpoints to intersection. 5. **Leave construction arcs visible** (examiner checks). ### Key Tips - Always draw a large, clear diagram. - Use sharp pencil, ruler, protractor, compasses. - For bearings, label North lines and angles clearly. - For scale, check unit conversions (bigger/smaller number sense).

Key terms

Bearing: An angle measured clockwise from North, written as three digits (e.g., 047°).
Scale: A ratio that compares the size of a drawing to the real-life size (e.g., 1 : 50 000 means 1 cm on drawing = 50 000 cm in reality).
SSS Triangle Construction: Drawing a triangle accurately given the lengths of all three sides, using a ruler and compasses, leaving construction arcs visible.
Reverse Bearing: If the bearing of A from B is less than 180°, add 180° to find the bearing of B from A; if greater than 180°, subtract 180°.
North Line: A reference line pointing north, drawn at the starting point when measuring or plotting bearings.

Slides

Bearings – The Basics

Bearings describe an angle used in navigation.
They are measured clockwise from North.
Always write bearings with 3 digits (e.g., 059°).

Which of the following is a correct bearing?

59°
059°
0059°
59.0°

Finding a Bearing

To find the bearing of A from B, start at B and measure clockwise from North to the line BA.
To find the bearing of B from A, start at A and measure clockwise from North to the line AB.
Always draw a North line at the starting point.

The bearing of A from B is measured clockwise from $____ at B$ .

Reverse Bearings

If the bearing of A from B is less than 180°, add 180° to get the bearing of B from A.
If the bearing of A from B is more than 180°, subtract 180° to get the bearing of B from A.

The bearing of A from B is 133°. What is the bearing of B from A?

047°
313°
227°
133°

Scale Drawings – Using a Scale

A scale is a ratio comparing drawn size to real size (e.g., 1:50000).
To find actual distance: measure on drawing, multiply by scale factor.
Convert units appropriately (e.g., cm to km).

On a map with scale 1:50000, 1 cm represents $____ km$ .

Constructing SSS Triangles

SSS means you know all three side lengths.
Draw the longest side as the base.
Use compasses to draw arcs from each end with the other side lengths; the intersection is the third vertex.
Do not erase construction arcs – they show your method.

Which statements about constructing an SSS triangle are true?

You should start by drawing the longest side.
You can use a protractor to measure angles.
Construction arcs must be left visible.
The compass width should be changed for each arc.

Drawing a Point on a Bearing

Step 1: Draw a North line at the starting point.
Step 2: Measure the given bearing clockwise from North.
Step 3: Draw a line in that direction and mark the point at the given distance.

To plot point Q from P on a bearing of 105°, you should measure the angle clockwise from which direction?

South
East
North
West

Scale – Converting Real Distances to Drawing

Convert the scale to a ratio like 1 cm : n km.
Divide the real distance by n to get the drawing distance in cm.
Example: scale 1 cm : 5 km, real 20 km → drawing 4 cm.

A map scale is 1:200000. A real distance of 10 km $is ____ cm$ on the map.

Constructing Triangles – Accuracy Tips

Use a sharp pencil for precise arcs and lines.
Check that the compass width is exactly the required length.
Label vertices and side lengths clearly.
Do not rub out arcs – they are part of your answer.

In an SSS triangle construction, why should you not erase the arcs?

They make the diagram look neat.
They show the examiner your method.
They help you measure angles.
They are required for the next step.

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