Constructions and loci Worked Solutions — GCSE Maths

Recall what a locus is

A locus is the set of every point that follows a given rule — here, every point at a fixed distance of $4$ cm from $P$.

Picture the compass point at P

Putting the compass point on $P$ and opening it to $4$ cm, one full sweep marks every point that is $4$ cm away. That sweep is a circle of radius $4$ cm.

Rule out the other descriptions

A radius of $8$ cm would be points $8$ cm away; the region inside the circle contains points closer than $4$ cm; and no straight line or single point keeps the distance fixed in every direction.

Test the midpoint first

The midpoint of $AB$ is certainly the same distance from both, so it belongs to the locus. But it is not the only such point.

Look for other equidistant points

Any point directly above or below the midpoint, on the line at right angles to $AB$, is also equidistant from $A$ and from $B$ by symmetry. Together these points form a whole straight line.

Name the locus

That line cuts $AB$ in half and meets it at right angles: it is the perpendicular bisector. It is not $AB$ itself, not a circle, and not a single point.

Recall the distance from a point to a line

The distance from a point to a fence means the perpendicular distance to it.

Find the points where the two distances are equal

Starting at $C$ and moving out so that both perpendicular distances grow at the same rate traces the line exactly halfway between the fences — the bisector of the angle at $C$. This is what the ruler-and-compass angle-bisector construction draws.

Rule out the other options

A line parallel to one fence keeps its distance from that fence but not from the other. A circle centred on $C$ has points at all sorts of distances from the fences, and a perpendicular to one fence is not halfway between them.

Fix the distance from a straight line

Every point of the locus is $3$ m from the hedge, measured at right angles to it.

Remember there are two sides

Points $3$ m away can be on either side of the hedge, so there are two parallel lines, not one.

Rule out the curved answers

The semicircular ends of a racetrack shape appear only when the line has ends. A long straight hedge is treated as a full straight line, so the locus is just the two parallel lines.

Recall why the midpoint matters

The perpendicular bisector crosses $AB$ exactly halfway along it, so the crossing point is the midpoint of $AB$.

Average the coordinates

The midpoint is found by averaging the $x$-coordinates and averaging the $y$-coordinates.

Work out the midpoint

This gives $M = (5, 3)$, which is the same distance from $A$ as from $B$.