What you'll learn
This topic covers ruler-and-compass constructions, loci and scale drawings. In this guide you will learn how to construct perpendicular bisectors, angle bisectors and perpendiculars, how to build up loci from a set of conditions, and how to combine constructions to solve "region" problems. Accurate use of compasses and a sharp pencil is essential throughout.
Key terms and definitions
Construction — an accurate drawing made with only a ruler and compasses.
Perpendicular bisector — a line that cuts another line in half at 90°.
Angle bisector — a line that cuts an angle exactly in half.
Locus — the set of all points that satisfy a given condition (plural: loci).
Scale drawing — an accurate diagram drawn to a stated scale.
Core concepts
Perpendicular bisector
To construct the perpendicular bisector of a line, open the compasses to more than half the length, draw arcs from each end above and below the line, and join where the arcs cross. Every point on this line is equidistant from the two ends.
Angle bisector
To construct an angle bisector, place the compass point on the vertex and draw an arc crossing both arms; from those two crossings draw arcs that meet, then join the vertex to that meeting point. Every point on the bisector is equidistant from the two arms.
Perpendicular from or to a point
You can drop a perpendicular from a point to a line (or erect one at a point on a line) using arcs to find two equidistant points, then bisecting. These give the shortest distance from a point to a line.
Loci
A locus is the path of all points obeying a rule. Common loci are: a fixed distance from a point (a circle), a fixed distance from a line (a "racetrack" with semicircular ends), equidistant from two points (the perpendicular bisector), and equidistant from two lines (the angle bisector). Build each from the matching construction.
Combining conditions for regions
Many problems ask for a region satisfying several conditions at once (e.g. within 5 cm of A and nearer to B than C). Draw each locus, then shade the overlap that meets all the conditions. Read "nearer to", "within" and "more than" carefully to shade the correct side.
Worked examples
Example 1: Equidistant from two points
Describe the locus of points equidistant from points A and B.
The perpendicular bisector of AB.
Example 2: Fixed distance from a point
Describe the locus of points exactly 4 cm from a point P.
A circle of radius 4 cm centred on P.
Example 3: Scale drawing
A plan uses a scale of 1 cm to 5 m. A wall is 8 cm on the plan. Find its real length.
8 × 5 = 40 m.
Common mistakes and how to avoid them
Rubbing out construction arcs. Leave them — they show your method and earn marks.
Compasses slipping. Keep the radius fixed and the point firmly placed.
Shading the wrong region. Re-read "within", "nearer to" and "more than".
Freehand "constructions". Use only ruler and compasses, not a protractor, when a construction is required.
Forgetting the scale. Convert between drawing and real distances using the stated scale.
Exam technique for Constructions and Loci
Keep all construction arcs visible.
Use the right construction for each locus condition.
Draw every locus before shading a region.
Shade the overlap that satisfies all conditions.
Work to the scale for scale-drawing measurements.
Quick revision summary
Constructions use only a ruler and compasses, and you should always leave the arcs showing. The perpendicular bisector (arcs from each end of a line) gives points equidistant from two points; the angle bisector (arcs from the vertex and arms) gives points equidistant from two lines. A locus is the set of points obeying a rule: a fixed distance from a point is a circle, a fixed distance from a line is a racetrack shape, equidistant from two points is the perpendicular bisector, and equidistant from two lines is the angle bisector. For region problems, draw every locus and shade the overlap that meets all conditions, reading "within", "nearer to" and "more than" carefully. Scale drawings convert between plan and real distances using the stated scale. The common errors are erasing arcs, slipping compasses, shading the wrong region, and ignoring the scale. Use the correct construction, keep the arcs, draw all loci, and shade the right overlap.