Scale Drawing
Geometry · Scale Drawing
Syllabus tag: Kenya CBC | Grade 9 Mathematics | Strand 4.0 Geometry | Sub-Strand 4.2 Scale Drawing (14 Lessons)
Lesson objectives
By the end of this sub-strand, you should be able to:
- Interpret and use scales expressed as ratios and as statements.
- Convert between drawing measurements and actual measurements.
- Determine compass and true bearings between points.
- Use scale drawings to solve practical problems involving distance, direction and area.
Scale Drawing
A surveyor cannot carry a field into the office. Instead they draw it small, but keep every angle and every proportion correct. That drawing is a scale drawing.
This topic is worth 14 lessons, the second longest in Grade 9 Mathematics. Most of the marks come from drawing carefully, not from calculating.
a) Compass directions
The compass gives you eight directions you already know.
North, East, South and West are the four main ones. Between each pair sits a direction named after both, such as North East.
Compass directions are useful for rough description. For accurate work we need something better.
b) Three-figure bearings
A bearing is an angle measured from North, turning clockwise, and always written with three figures.
Two rules make bearings easy to get right.
Always start at North, never anywhere else. Always turn clockwise, never anticlockwise.
Write 060°, not 60°. The extra zero is not decoration. It tells the reader you have given a bearing.
A bearing can be anything from 000° up to 360°. South West, for example, is 225°.
c) The bearing of one point from another
Read the question carefully. "The bearing of B from A" means you stand at A and look towards B. Standing at the wrong point is the most common mistake in this topic.
If you swap the two points, the bearing changes by 180°.
If the first bearing is more than 180°, subtract 180 instead of adding it. Your answer must always land between 000° and 360°.
d) Locating a point using bearing and distance
Two pieces of information fix a point exactly: which direction, and how far.
To draw it, work in this order. Mark your starting point. Draw a light North line through it. Measure the bearing clockwise from that North line with a protractor. Then measure the distance along it with a ruler, using your scale.
Keep your scale written on the drawing. A drawing without its scale cannot be read back.
e) Angles of elevation
When you look up at something, the angle between your line of sight and the level ground is the angle of elevation.
To find it by scale drawing, measure the distance to the base of the object and its height. Draw both to scale. Then measure the angle at your position with a protractor.
f) Angles of depression
When you look down at something, the angle between your line of sight and the level line is the angle of depression.
These two angles are closely linked. Look down from the cliff to the boat, and look up from the boat to the cliff. Both give the same angle. It is one line of sight, measured from either end.
g) Simple surveying
Surveying means fixing the positions of several points and drawing them to scale.
Work from one fixed point, called a base. Take the bearing and distance to each other point from that base. Plot them one at a time. Once every point is on the paper, read any distance or angle straight off the drawing.
h) Where this is used
Surveyors setting out plot boundaries in Kajiado use bearings every day. So do pilots and ship navigators. A county officer planning a water pipeline scale-draws the route first. That tells them how much pipe to order before anyone digs.
Words to know
- Scale -- the fixed ratio between a length on a drawing and the corresponding real length.
- Scale factor -- the number by which a drawing length is multiplied to give the actual length.
- True bearing -- a direction measured clockwise from north, written with three digits.
- Compass bearing -- a direction stated as an angle east or west of north or south.
- Back bearing -- the reverse direction, differing from the original by 180 degrees.
:::checkpoint Check yourself
- Write South East as a three-figure bearing.
- The bearing of Q from P is 125°. What is the bearing of P from Q?
- On a map with scale 1 : 25 000, two towns are 8 cm apart. How far apart are they in kilometres?
- You stand on a bridge and look down at a boat. Is that an angle of elevation or an angle of depression? :::
Bridge to practice
The exercises begin with reading scales and converting distances, move through compass and true bearings and back bearings, and finish with area scaling and a two-leg journey problem. Where a question describes a journey, sketch it before choosing an option; even a rough sketch usually makes the correct answer obvious.