Construct triangles, angle bisectors and perpendicular bisectors with compasses, and describe and draw simple loci

What this dot point is asking

SEAB wants you to use compasses and a straight edge to construct triangles and bisectors accurately, and to describe and draw loci, the set of all points satisfying a given condition. Loci questions often combine two conditions and ask for the region where both hold.

The answer

Constructing a triangle

Given three sides, draw the base with a ruler, then use compasses set to each of the other two lengths to draw arcs from the two ends; their intersection is the third vertex. Always leave your construction arcs visible, as they show the method and earn marks.

Perpendicular bisector

The perpendicular bisector of a segment $AB$ is constructed by opening the compasses to more than half of $AB$ and drawing arcs from both $A$ and $B$ above and below the line; the line through the two arc intersections is the perpendicular bisector. Every point on it is equidistant from $A$ and $B$.

Angle bisector

To bisect an angle, draw an arc from the vertex crossing both arms, then from those two crossing points draw two more arcs that meet; the line from the vertex through that meeting point bisects the angle. Every point on it is equidistant from the two arms.

Standard loci

• The locus of points a fixed distance $r$ from a point is a circle of radius $r$.
• The locus of points equidistant from two points is the perpendicular bisector of the segment joining them.
• The locus of points equidistant from two lines (or the arms of an angle) is the angle bisector.
• The locus of points a fixed distance from a line is a pair of parallel lines.

Combining loci

When a point must satisfy two conditions at once, draw each locus and identify the overlap, the region or points common to both. Less than a distance gives the inside of a circle; nearer to one side gives one side of a bisector.

Examples in context

Example 1. Positioning a transmitter. A radio mast that must be equidistant from two towns sits on the perpendicular bisector of the line joining them. Adding a maximum-range condition (a circle) pins it to part of that bisector.

Example 2. Safe zones. A path that must stay at least a set distance from a building edge is bounded by a locus parallel to the edge. Planners use such loci to mark out clearances and safety margins.

Try this

Q1. Describe the locus of points $5\ \text{cm}$ from a fixed point $P$. [1 mark]

• Cue. A circle of radius $5\ \text{cm}$ centred on $P$.

Q2. What construction gives the locus of points equidistant from two intersecting lines? [1 mark]

• Cue. The angle bisector of the angle between them.

Q3. A point must be within $3\ \text{cm}$ of $A$ and equidistant from $A$ and $B$. Describe the locus. [2 marks]

• Cue. The part of the perpendicular bisector of $AB$ that lies inside the circle of radius $3\ \text{cm}$ centred on $A$.