01 — Explanation
What is a locus?
A locus is the complete set of points satisfying one specific geometric rule — for example, "every point exactly 3 cm from P," or "every point equidistant from two lines." Constructions let you draw these rules precisely using only a ruler and a pair of compasses — never a protractor, except for measuring bearings.
Exam Tip
Bring compasses that hold their setting (not "loose and wobbly") and a sharp pencil — physically unstable equipment is a common, entirely avoidable reason for losing accuracy marks on construction questions.
02 — Explanation
How do I construct the perpendicular bisector of a line (the locus equidistant from two points)?
- Open the compasses to more than half the length of the line segment AB.
- From A, draw an arc above and below the line.
- Keeping the same radius, repeat from B so the arcs cross the first pair.
- Join the two intersection points with a straight line — this line is perpendicular to AB, passes through its midpoint, and every point on it is equidistant from A and B.
03 — Explanation
How do I construct the angle bisector (the locus equidistant from two lines)?
- From the vertex, draw an arc crossing both lines of the angle.
- From each of those two crossing points, draw a new arc inside the angle, using the same compass width both times.
- Join the vertex to where these two interior arcs intersect — this line bisects the angle exactly.
Exam Tip
Changing the compass width partway through either construction is the single most common procedural error — it invalidates the geometry, not just the neatness.
04 — Explanation
What are the four loci rules I need to recognise?
| Verbal condition | Locus (the boundary shape) |
|---|---|
| Fixed distance from a point | A circle, radius , centred on the point |
| Fixed distance from a line segment | Two parallel lines distance away, capped with semicircular arcs at each end (a "capsule" shape) |
| Equidistant from two points | The perpendicular bisector of the segment joining them |
| Equidistant from two intersecting lines | The angle bisector of the angle between them |
05 — Explanation
How do I shade a region defined by multiple loci at once?
Construct each boundary separately, then identify the overlap that satisfies all conditions simultaneously. For "closer to A than B," shade the side of the perpendicular bisector that contains A (not the line itself, which is where the distances are equal). For "within 3 cm of P," shade the inside of the circle.
Exam Tip
Shading the wrong side of a perpendicular bisector is the most common conceptual error — always double check which point ("closer to...") you were asked about before shading.
06 — Explanation
Why must I never rub out my construction arcs?
The compass arcs are the physical evidence that you used the proper method rather than estimating by eye — they are worth Method marks in their own right. A perfectly drawn final line with no supporting arcs can lose marks even though the answer looks correct; draw them lightly, but never erase them.
07 — Worked example
Worked example
A garden is a rectangular field. A tree is to be planted so that it is equidistant from two fence corners A and B, and also within 5 m of a third corner C. Describe how to find all the valid positions for the tree.
- Construct the perpendicular bisector of AB (equidistant from A and B) using the compass method above, keeping both arcs' radii identical and greater than half of .
- Construct a circle of radius 5 m centred on C.
- The valid planting positions are every point where the perpendicular bisector line passes inside the circle — mark this section of the line clearly as your final locus.