Bearings Constructions And Scale Drawings

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Notes

Bearings

**Bearings** are a way of describing an angle, commonly used in navigation.
Three rules: measured from **North**, measured **clockwise**, written with **3 digits** (e.g., 059°).
Compass directions have standard bearings: $N=000^{\circ}, E=090^{\circ}, S=180^{\circ}, W=270^{\circ}$ .
To find bearing of B from A: start at A, draw North line, measure clockwise angle to B.
If bearing of A from B $is <180^{\circ}$ , add 180° to get bearing of B from A; $if >180^{\circ}$ , subtract 180°.
Always draw a clear diagram and label angles carefully.

Scale

**Scale** is a ratio comparing drawn size to real-life size (e.g., 1:50000).
To find actual distance: measure on drawing, multiply by scale factor, convert units.
To find drawing length: convert actual distance to same units, divide by scale factor.
Common conversions: 1 $km = 1000 m = 100000$ cm.
Always check if answer is sensible (e.g., actual distance > drawing distance).

Constructing Triangles (SSS)

**SSS triangle** construction uses three given side lengths.
Tools: pencil, ruler, compasses.
Step 1: Draw the longest side as base using ruler.
Step 2: Set compass to one side length, draw arc from one endpoint above base.
Step 3: Set compass to third side length, draw arc from other endpoint to intersect first arc.
Step 4: Draw lines from base endpoints to intersection point.
Do **not erase construction arcs** – they show your method.
Label vertices and side lengths as required.

Scale Drawings

**Scale drawing** uses a given scale to represent real objects or distances.
Use ruler and protractor for accurate measurements and bearings.
To mark a point given distance and bearing: draw North line, measure bearing clockwise, draw line to scale distance.
For combined conditions (e.g., within distance range and bearing range), construct and shade the region.
Always leave construction lines and arcs visible for marks.

Bearing Measurement

Scale Conversion

SSS Triangle Construction

Scale Drawing with Bearing

Practice questions

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1.What are the three rules for bearings?
Easy
- AMeasured from North, clockwise, written with three digits
- BMeasured from South, anticlockwise, written with three digits
- CMeasured from East, clockwise, written with two digits
- DMeasured from North, anticlockwise, written with three digits
2.The bearing of B from A is 105°. What is the bearing of A from B?
Easy
- A285°
- B075°
- C255°
- D195°
3.A map has a scale of 1:50000. What does 1 cm on the map represent in real life?
Easy
- A50000 cm
- B5000 cm
- C500 m
- D5 km
4.In a scale drawing, 1 cm represents 40 cm. If the window on the drawing is 3 cm long and 2 cm high, what are the actual dimensions?
Easy
- A120 cm by 80 cm
- B40 cm by 30 cm
- C12 cm by 8 cm
- D160 cm by 120 cm
5.Using a ruler and compasses only, to construct a triangle with sides 5 cm, 7 cm and 9 cm, which side should you draw first?
Easy
- AThe longest side (9 cm)
- BThe shortest side (5 cm)
- CAny side, it doesn't matter
- DThe side of length 7 cm
6.What is the compass direction for a bearing of 090°?
Easy
- AEast
- BNorth
- CWest
- DSouth
7.The bearing of a lighthouse from a boat is 135°. What is the bearing of the boat from the lighthouse?
Medium
- A315°
- B045°
- C225°
- D135°
8.On a map with scale 1:150000, the distance between two towns is 5.8 cm. What is the actual distance in km?
Medium
- A8.7 km
- B87 km
- C0.87 km
- D870 km

Summary

## Bearings, Constructions & Scale Drawings ### Bearings - **Definition**: A bearing is an angle measured clockwise from North, written as three digits (e.g., 059°). - **Rules**: 1. Measured from North. 2. Measured clockwise. 3. Always written with 3 figures. - **Finding a bearing**: Start at the point after "from". Draw a North line, measure clockwise to the line joining the points. - **Reverse bearing**: If bearing of A from B is less than 180°, add 180°; if more than 180°, subtract 180°. ### Scale - **Definition**: A ratio comparing drawn size to real size. E.g., 1:50000 means 1 cm on map = 50000 cm in real life. - **Using scale to find actual length**: Measure on map, multiply by scale factor, convert units. - **Using scale to find drawn length**: Convert actual distance to same units, divide by scale factor. ### Constructing Triangles (SSS) - **Equipment**: Pencil, ruler, compasses. - **Steps**: 1. Draw the longest side as base. 2. Set compass to one side length, draw arc from one endpoint. 3. Set compass to third side, draw arc from other endpoint to intersect. 4. Connect endpoints to intersection. 5. **Do not erase construction arcs** (they earn marks). ### Tips - Always draw a clear diagram. - Use a sharp pencil and accurate ruler/protractor. - For bearings, label North lines.

Key terms

Bearing: An angle measured clockwise from North, written with three digits (e.g., 059°).
Scale: A ratio that describes the relationship between the drawn size and the real-life size (e.g., 1:50000).
SSS Triangle Construction: Drawing a triangle accurately given the lengths of its three sides using a ruler and compasses.
Construction Arcs: Arcs drawn with compasses during construction; must not be erased as they show method.

Slides

Bearings: The Basics

Bearings describe an angle used in navigation.
They are measured clockwise from North.
Bearings are always written with three digits, e.g. 059°.
North is usually straight up on a scale drawing.

Which of the following is a correct bearing?

59°
059°
0059°
59.0°

Finding a Bearing Between Two Points

Start at the point 'from' which the bearing is measured.
Draw a North line at the starting point.
Draw a line to the other point.
Measure the clockwise angle from North to that line.

To find the bearing of B from A, where do you start?

At B
At A
At North
At the midpoint

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°.
Example: Bearing of A from $B = 073^{\circ}$ , then bearing of B from $A = 073^{\circ} + 180^{\circ} = 253^{\circ}$ .

The bearing of A from B is 105°. What is the bearing of B from A?

285°
075°
195°
255°

Scale Drawings: Using a Scale

A scale is a ratio, e.g. 1:50000 means 1 cm on the drawing $= 50000$ cm in real life.
To find actual distance: measure on drawing, multiply by scale factor.
Convert units appropriately (e.g. cm to km).

A map scale is 1:50000. A distance on the map is 4 cm. What is the actual distance in km?

2 km
20 km
0.2 km
200 km

Constructing SSS Triangles

You need a ruler, compasses, and pencil.
Draw the longest side as the base.
Use compasses to draw arcs from each end with the other two side lengths.
The intersection of arcs is the third vertex. Do not erase arcs.

When constructing an SSS triangle, which side should you draw first?

The shortest side
The longest side
Any side
The side with the middle length

Drawing a Point on a Bearing

Draw a North line at the starting point.
Measure the bearing clockwise from North.
Draw a line in that direction.
Use the scale to mark the distance along that line.

You need to mark point B from A on a bearing of 140° at a distance of 36 km. Scale: 1 $cm = 4$ km. How many cm from A should you measure?

9 cm
4 cm
36 cm
140 cm

Scale as a Ratio

A scale can be written as 1:n, e.g. 1:12000.
Convert both parts to the same units before simplifying.
Example: 5 cm represents 0.6 km → 5 cm : 60000 cm → 1:12000.

A map uses 1 cm to represent 0.5 km. What is the scale as 1:n?

1:500
1:5000
1:50000
1:500000

Constructing Equilateral Triangles

An equilateral triangle has all sides equal.
Given base AB, set compasses to length AB.
Draw arcs from A and B above the base.
The intersection is the third vertex C.

To construct an equilateral triangle with base AB, what compass setting do you use?

Any length
The length of AB
Half of AB
Twice AB

Measuring Bearings with a Protractor

Place the protractor's centre at the starting point.
Align 0° with North.
Read the clockwise angle to the line to the other point.
Write the angle as three digits.

When measuring a bearing, you should measure the angle:

Clockwise from North
Anticlockwise from North
Clockwise from South
From the nearest compass point

Flashcards

What are the three rules for bearings?

Measured from North, measured clockwise, and written with three digits (e.g., 059°).

How do you find the bearing of B from A?

Start at A, draw a North line, measure the clockwise angle from North to the line AB, and write it as three digits.

If the bearing of A from B is 105°, what is the bearing of B from A?

Add $180^{\circ}: 105^{\circ} + 180^{\circ} = 285^{\circ}$ .

If the bearing of A from B is 245°, what is the bearing of B from A?

Subtract $180^{\circ}: 245^{\circ} - 180^{\circ} = 065^{\circ}$ .

What is the bearing of East?

090°.

What is the bearing of South?

180°.

What is the bearing of West?

270°.

What is the bearing of North?

000° or 360°.

A map scale is 1 cm represents 10 km. What is the scale as a ratio?

1 : 1 000 000 (since 10 $km = 1 000 000 cm)$ .

On a map with scale 1:50 000, a distance measures 5.8 cm. What is the actual distance in km?

5.8 $cm \times 50 000 = 290 000$ $cm = 2.9$ km.

How do you construct an SSS triangle with sides 6 cm, 7 cm, and 10 cm?

Draw the 10 cm base. Using compasses, draw an arc of radius 6 cm from one end and an arc of radius 7 cm from the other end. The intersection is the third vertex. Connect and label sides.

Why should you not erase construction arcs in triangle constructions?

Examiners use the arcs to verify that the triangle was constructed accurately.

How do you construct an equilateral triangle with side AB given?

Set compasses to length AB. Draw arcs from A and B above AB. The intersection is C. Connect A to C and B to C.

A scale drawing uses 1 cm to represent 40 cm. The window on the drawing is 3 cm long and 2 cm high. Find actual length and height.

Length $= 3 \times 40 = 120$ cm; Height $= 2 \times 40 = 80$ cm.

On a map, 1 cm represents 15 km. The distance between two towns on the map is 4.5 cm. Find actual distance.

$4.5 \times 15 = 67.5$ km.

How do you mark a point C on a scale drawing given its distance and bearing from A?

Draw a North line at A. Measure the bearing clockwise from North. Draw a line in that direction. Measure the scaled distance along that line to mark C.

The bearing of B from A is 073°. Find the bearing of A from B.

$073^{\circ} + 180^{\circ} = 253^{\circ}$ .

What equipment is needed for SSS triangle construction?

Pencil, ruler, and a pair of compasses.

A map scale is 1:500 000. Convert this to 1 cm represents how many km?

1 cm represents 5 km (since 500 000 $cm = 5 km)$ .

How do you find the actual distance from a scale drawing if the scale is given as a ratio?

Measure the distance on the drawing in cm, multiply by the scale factor, then convert to suitable units.

In triangle $ABC, BC = 7.6$ cm and $AC = 6.2$ cm. AB is drawn. How do you construct the triangle?

Set compasses to 6.2 cm, draw an arc from A. Set to 7.6 cm, draw an arc from B. Intersection is C. Connect A-C and B-C.

A field ABC has $AC = 500 m$ and $BC = 650 m$ . Scale 1 $cm = 100 m$ . AB is drawn. How to complete the scale drawing?

Convert: $AC = 5$ cm, $BC = 6.5$ cm. Draw arcs of these radii from A and B to find C.

What is the bearing of a point directly west of another?

270°.

A circular garden has diameter 11.4 m. Scale 1 $cm = 1.5 m$ . What is the diameter on the drawing?

$11.4 \div 1.5 = 7.6$ cm.

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