17.4 Loci — Learn

Prerequisites: construct a perpendicular bisector and draw a circle with compasses.

Objectives: describe and construct loci, combine conditions, and use the shortest distance from a point to a line. This page covers the loci and shortest-distance parts of G2.

What is a locus?

A locus is the set of all points that satisfy a stated condition. One point can satisfy several conditions; the answer is where their loci overlap.

• Points a fixed distance from a point form a circle.
• Points a fixed distance from a straight line form two parallel lines, one on each side.
• Points equally distant from two points lie on the perpendicular bisector of the segment joining them.
• Points equally distant from two intersecting lines lie on the angle bisectors.

Worked example. Mark points exactly $4$ cm from point $P$.

• Every point at distance $4$ cm from $P$ lies on a circle centred at $P$.
• Draw the circle with compasses set to $4$ cm.
• Any point on that circumference meets the condition.

Distance from a point to a line

The perpendicular distance is the shortest distance from a point to a line. A perpendicular makes a right angle with the line.

Worked example. Point $Q$ is $3$ cm above a horizontal line. Compare a sloping route to the line with a perpendicular route.

• Every sloping route forms the hypotenuse of a right triangle with vertical height $3$ cm.
• The hypotenuse is longer than either shorter side.
• Therefore the perpendicular route, of length $3$ cm, is shortest.

Watch out. Measuring along a sloping segment does not give the distance to a line. The distance is measured at $90^\circ$; draw a right-angle mark to check.

Progress check

The circular bed is exactly $4$ m from the water point. Describe and sketch the locus of the bed boundary. Then explain why a point $3$ m from the water point is not on the boundary.

You should now be able to

• describe the locus for a fixed distance from a point or line;
• construct loci for equal distances from points or lines;
• find intersections that satisfy two conditions;
• explain why perpendicular distance is shortest.