Constructions and Loci
A construction uses geometric facts, not just measurement. Compass arcs are part of the reasoning, so they should normally remain visible. Loci then use these same ideas to describe all points satisfying a condition.
Foundation & Higher. This lesson builds on basic angle and shape language and links to Units, Scale Drawings and Maps when a real-world scale is involved.
Perpendicular bisector
To construct the perpendicular bisector of AB: set a compass radius greater than half of AB, draw equal-radius arcs from A and B above and below the segment, then join the two arc intersections.
Every point on this line is equidistant from A and B.
Perpendicular through or from a point
A perpendicular can be constructed at a point on a line or from an external point to a line. In each case, the construction creates two points on the line that are equidistant from the chosen point, then bisects between them.
The perpendicular gives the shortest distance from a point to a line.
Perpendicular from an external point
Let P lie outside a line. Centre the compasses at P and draw an arc crossing the line at A and B. Using equal compass widths from A and B, draw arcs that meet at Q on the other side of the line. Draw PQ. Since PA=PB and QA=QB, both P and Q lie on the perpendicular bisector of AB, so PQ is perpendicular to the line.
Perpendicular at a point on the line
Let P be on the line. With centre P, mark A and B at equal distances on opposite sides of P. Draw equal-radius arcs from A and B, with radius greater than AP, to locate Q off the line. Join P to Q. Leave the arcs visible; the symmetry makes the angle at P exactly 90°.
Angle bisector and 60° construction
To bisect an angle, draw an arc from the vertex to hit both arms, then draw equal arcs from those two hit points. Join the vertex to the new arc intersection.
To construct 60°, build an equilateral triangle: with a fixed compass radius, draw arcs from the endpoints of a segment. Joining to an intersection creates three equal sides and therefore 60° angles. Bisection can then create 30°.
The four core loci
| Condition | Locus |
|---|---|
| Fixed distance from a point | Circle centred at that point |
| Fixed distance from a straight line | Two parallel lines, one on each side |
| Equidistant from two points | Perpendicular bisector |
| Equidistant from two intersecting lines | The two angle bisectors |
If the reference is a finite line segment rather than an infinite line, the locus at a fixed distance includes semicircular end effects.
Combining conditions
Words matter: and means the overlap of all conditions; or means points satisfying either condition. Draw each boundary first, decide whether it is included, then shade only the region that survives every required condition.
Example: ‘within 4 cm of A and closer to B than C’ means inside the circle centred A and on the B side of the perpendicular bisector of BC.
Think Like a Mathematician
A locus is not one guessed point. It is the complete set of all points that satisfy a condition. Construction lines reveal why that set has its particular shape.
Common Mistakes
Erasing construction arcs: the arcs show that the construction is genuine.
Using a protractor instead of the required compass construction.
Confusing an angle bisector with a perpendicular bisector.
Shading the union when the question says ‘and’.
Practice — constructions and loci
1. What line contains all points equidistant from points A and B?
The perpendicular bisector of AB.
2. What is the locus of points exactly 5 cm from point P?
A circle centred at P with radius 5 cm.
3. What is the locus of points exactly 2 cm from an infinite straight line?
Two lines parallel to the original, each 2 cm away.
4. Why must the compass radius exceed half of AB when constructing a perpendicular bisector?
So the equal-radius arcs from A and B intersect in two places.
5. State the shortest distance from a point to a line in geometric terms.
The length of the perpendicular from the point to the line.
6. How can a 30° angle be constructed from an equilateral-triangle construction?
Construct 60° first, then bisect it.
7. What is the locus of points equidistant from two intersecting lines?
The two angle bisectors.
8. Draw P and Q 5 cm apart. Construct and shade the region within 3 cm of P and within 4 cm of Q.
Draw circles centred at P and Q with radii 3 cm and 4 cm. Shade their lens-shaped overlap, including the boundaries: every shaded point satisfies both distance conditions.
9. A region must be closer to A than B. Which side of the perpendicular bisector of AB is used?
The half-plane on the A side of the perpendicular bisector.
10. Why can a finite line segment create rounded end effects in a fixed-distance locus?
Near an endpoint, points at the fixed distance form part of a circle centred on that endpoint.
11. Draw AB=8 cm. Use ruler and compasses to construct its perpendicular bisector. Mark the two points that are 5 cm from both A and B.
Draw equal arcs of radius greater than 4 cm from A and B and join their two intersections: this line crosses AB at its midpoint and at 90°. Then draw circles of radius 5 cm centred at A and B. Their two intersections are the required points; both lie on the bisector. Keep your arcs visible.
12. If a question says ‘at least 2 cm from a line’, which side of each parallel boundary is allowed?
The regions outside the 2 cm strip around the line, including the boundaries if ‘at least’ includes equality.
Frequently Asked Questions
Can I use a protractor for a construction question?
Only if the question permits measuring. A ruler-and-compass construction must use the stated instruments.
Should I erase arcs?
Usually no. The arcs provide evidence of the construction method.
Does a locus boundary count?
It depends on the wording: ‘at most’ or ‘within’ includes the boundary; ‘less than’ excludes it.