Constructions, Scale Drawings and Loci
This is the most practical topic in the syllabus — the marks are for accurate drawing with the right equipment, not for calculation. The rule that governs everything: leave your construction arcs visible.
1. Equipment and the golden rules
You need a pair of compasses, a ruler, a protractor and a sharp pencil.
Use a pencil, never a pen. You will need to correct something, and pen cannot be erased — this was a recorded error.
DO NOT RUB OUT YOUR ARCS. The construction arcs are the evidence that you used compasses rather than guessing, and marks are awarded for them. Tutors were emphatic: an accurate-looking line with no arcs can score nothing.
Use compasses for constructions, not a ruler and estimation. “Construct” is a technical instruction meaning compasses-and-straightedge.
Label points and lengths clearly — another documented loss.
Accuracy tolerance is usually ±2 mm for lengths and ±2° for angles, so work carefully with a sharp pencil.
2. Constructing a triangle
Given three sides (SSS):
- Draw the longest side as the base with a ruler
- Set compasses to the second side’s length; draw an arc from one end
- Set compasses to the third side’s length; draw an arc from the other end
- The arcs cross at the third vertex
- Join the vertices — leaving the arcs
The intersection of the two arcs gives the vertex. Drawing arcs that don’t cross, or crossing the wrong pair, was a recorded difficulty — make your arcs long enough to intersect clearly.
Given two sides and the included angle (SAS): draw one side, measure the angle with a protractor, draw the second side, then join.
Start measuring from ZERO on the protractor, and use the correct scale — protractors have two, and picking the wrong one gives 180° minus your angle. This was specifically flagged in lessons.
3. The two standard constructions
Perpendicular bisector of a line AB
- Set compasses to more than half of AB
- Draw arcs above and below from A
- Without changing the compass width, draw arcs from B
- Join the two intersection points
This line is perpendicular to AB and passes through its midpoint.
Every point on the perpendicular bisector is EQUIDISTANT from A and B. That is what makes it a locus, and it is the property questions test.
Do not change the compass setting between steps 2 and 3.
Angle bisector
- Put the compass point on the vertex; draw an arc crossing both arms
- From each crossing point, draw arcs that intersect
- Join the vertex to that intersection
Every point on the angle bisector is EQUIDISTANT from the two lines.
4. Loci
A locus is the set of all points satisfying a given condition. (Plural: loci.)
The four standard loci:
| Condition | Locus |
|---|---|
| A fixed distance from a point | a circle |
| A fixed distance from a line | two parallel lines either side, with semicircular ends |
| Equidistant from two points | the perpendicular bisector |
| Equidistant from two lines | the angle bisector |
Method for a region question:
- Draw each locus separately, as a construction
- Decide which side of each satisfies the condition
- Shade the region satisfying all conditions
- Label it
“Less than 5 cm from A” is the INSIDE of the circle; “more than” is the outside. Read each condition and mark the correct side before shading.
Use a solid line if the boundary is included, and shade only where all the conditions overlap. Questions typically combine two or three.
5. Scale drawings
A scale of 1 : 200 means 1 cm on the drawing represents 200 cm in reality.
Method:
- Convert the real distances to drawing lengths — divide by the scale
- Draw accurately with a ruler and protractor
- Measure what the question asks
- Convert back to real life — multiply by the scale
State the units of your final answer, and convert sensibly — 20 000 cm should be given as 200 m.
Bearings frequently appear here: measure clockwise from north and give three figures.
Estimating area by counting squares: count whole squares, then count part-squares as half each (or count those more than half full). This gives an estimate, which is all that is asked.
6. Mistakes that cost marks
Rubbing out the construction arcs.
Using a ruler instead of compasses for a construction.
Using pen.
Changing the compass width part-way through.
Reading the wrong scale on the protractor, or not starting from zero.
Drawing inaccurately, outside the tolerance.
Not labelling points, lengths or the required region.
Shading the wrong side of a locus.
Shading where only some conditions are met.
Forgetting to convert with the scale, or omitting units.
Frequently asked questions
What is a locus? The set of all points satisfying a given condition.
What is the locus of points equidistant from two points? The perpendicular bisector of the line joining them.
What is the locus of points equidistant from two lines? The angle bisector.
What is the locus of points a fixed distance from a point? A circle.
How do I construct a perpendicular bisector? Equal arcs from both ends of the line, above and below, then join the intersections.
Why must I keep my arcs? They are the evidence of construction, and marks are given for them.
What equipment do I need? Compasses, ruler, protractor and a sharp pencil.
How accurate must I be? Usually within 2 mm and 2°.
How do I use a scale of 1 : 200? Divide real lengths by 200 to draw; multiply drawing lengths by 200 to convert back.
Quick revision checklist
- I use compasses, ruler, protractor and pencil
- I never rub out my arcs
- I can construct a triangle from three sides
- I can construct a triangle from two sides and an angle
- I start protractor measurements from zero, on the right scale
- I can construct a perpendicular bisector
- I can construct an angle bisector
- I know the equidistance property of each
- I know the four standard loci
- I can shade a region satisfying several conditions
- I label points, lengths and regions
- I can work with scale drawings in both directions
- I give units on real-life answers
These notes cover constructions, scale drawings and loci in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons. This was one of the least-covered topics in that set, so the page follows the syllabus closely rather than being padded — the flagged errors are drawn from the lessons that did cover it. Always check the current syllabus for your own exam series.
Finished this topic?
Saved on this device — no account needed.