Scale Drawings and Bearings
From navigating with a map to reading a builder’s plans, scale drawings and bearings are essential skills for understanding the world around us. This guide shows you how to measure lengths and angles accurately, interpret scale, and master the art of bearings.
The Core Concepts
Scale Drawings
A scale drawing represents a real object with all dimensions proportionally reduced or enlarged. The scale tells you the relationship between the drawing and real life.
- Ratio scale (e.g., $1:100$): 1 unit on the drawing $=$ 100 of the same units in real life.
- Statement scale (e.g., $1\text{ cm} = 5\text{ m}$): 1 cm on the drawing represents 5 metres in real life.
Real life → Drawing: Divide by the scale factor
Bearings
Bearings describe direction precisely. Three rules — always:
- Measure from North.
- Measure clockwise.
- Write as a three-figure number (e.g., $045°$, $270°$).
Worked Examples
Example 1: Using a Scale
A map has a scale of $1:50{,}000$. The distance between two towns on the map is 8 cm. What is the real distance in km?
- Find real distance in cm:
$8\text{ cm} \times 50{,}000 = 400{,}000\text{ cm}$. - Convert cm to km:
$400{,}000 \div 100{,}000 = 4\text{ km}$.
Answer: 4 km
Example 2: Measuring a Bearing
Find the bearing of point B from point A.
- Draw a North line at A (the “from” point).
- Place a protractor at A with 0° on the North line.
- Measure clockwise to the line AB.
Answer: bearing of B from A = 220°
Tutor Insights
🤔 Common Misunderstandings
- Scale direction: Mixing up when to multiply and when to divide. Remember — drawing to real life: multiply; real life to drawing: divide.
- Bearing direction: Forgetting that bearings must be measured clockwise from North, not anticlockwise.
- The “from” point: Always draw the North line at the point you are measuring from, not the destination.
📝 Common Exam Mistakes
- Forgetting the three-figure rule for bearings — write 045°, not 45°.
- Inaccurate measurements — aim to be within 1 mm for lengths and 1° for angles.
- Unit conversion errors, especially forgetting that $1\text{ km} = 100{,}000\text{ cm}$.
- Back-bearings: The bearing of A from B differs from the bearing of B from A by exactly 180°.
Practice Questions
- A map has a scale of $1\text{ cm}$ represents $2\text{ km}$. If two villages are 5 cm apart on the map, what is the actual distance between them?
- On a diagram, a line measures $7.5\text{ cm}$. The scale is $1:200$. What is the actual length of the line in metres?
- A school field is a rectangle measuring 80 m by 50 m. Draw a scale diagram of the field using a scale of $1\text{ cm} = 10\text{ m}$. What are the dimensions of the rectangle on your drawing?
- A ship sails from port P on a bearing of 065°. After reaching point Q, it turns and sails to point R on a bearing of 155°. Describe the journey from Q back to P using a back-bearing.
- Point A is due North of point B. What is: (a) the bearing of B from A? (b) the bearing of A from B?
Show Answers
- $5\text{ cm} \times 2\text{ km/cm} = \mathbf{10\text{ km}}$.
- $7.5 \times 200 = 1500\text{ cm} \div 100 = \mathbf{15\text{ m}}$.
- Drawn length $= 80 \div 10 = 8\text{ cm}$. Drawn width $= 50 \div 10 = 5\text{ cm}$. Answer: a rectangle 8 cm by 5 cm.
- The back-bearing from Q to P = $065° + 180° = \mathbf{245°}$.
- (a) B is due South of A. South $= \mathbf{180°}$. (b) A is due North of B. North $= \mathbf{000°}$.
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