Loci and Regions

Turning a rule into a shape

Stand in a field and obey one simple order: “stay exactly 3 metres from this post.” Walk while keeping that rule and your footprints trace a perfect circle. Change the order to “stay the same distance from these two posts” and your path becomes a dead-straight line running down the middle. You never chose the shape — the rule chose it for you.

That path is a locus (say “LOW-kuss”; the plural is loci, “LOW-sigh”). A locus is the set of all the points that obey a rule about distance — and only those points. Loci are how we turn a condition (“near enough”, “equally far”, “out of range”) into a shape you can draw: safe zones, boundaries, reachable areas, the patch of ground a sprinkler wets, the strip a robot arm can touch.

The four loci you must know

Almost every locus question at this level is built from just four standard results. Learn these and you can read most problems on sight:

points a fixed distance from a single point form a circle (centre that point, radius the fixed distance);
points a fixed distance from a line form a pair of parallel lines, one on each side — joined at the ends by semicircle caps if the line is really a finite segment (think of the rounded outline around a running-track lane);
points equidistant from two points form the perpendicular bisector of the segment joining them;
points equidistant from two lines form the angle bisector of the angle between them.

The first two are about distance from one thing; the last two are about being equally far from two things. Spot which kind you have and the locus almost draws itself.

Loci, and regions built from them

A locus is usually a line or curve — a one-pixel-thick boundary. But often you want a whole region: not “exactly 3 cm from A” but “within 3 cm of A”, the filled-in disc. Stack two or more conditions together and the region is the part of the plane where all of them hold at once — the overlap.

fixed distance from a point → a circle;
fixed distance from a line → a pair of parallel lines;
equidistant from two points → the perpendicular bisector;
equidistant from two lines → the angle bisector;
a region (e.g. “within 3 cm and nearer A than B”) is the overlap of the loci — shade only where all the conditions hold.

See two loci

Step through the figure: first the locus of points a fixed distance from one point P, then the locus of points equidistant from two points A and B.

Worked example 1 — draw a single locus

“A treasure is buried exactly 5 m from an old oak tree. Draw the locus of possible burial spots.”

The rule is a fixed distance (5 m) from a single point (the tree).
That is the first standard locus: a circle.
Put the compass point on the tree, open it to 5 m at the map’s scale, and draw the full circle. Every point on it is a candidate; nothing inside or outside qualifies.

Notice the word exactly: the answer is the circle, a curve. If the clue had said “within 5 m” the answer would instead be the whole disc — the shaded inside of the circle.

Worked example 2 — shade a two-condition region

“Shade the region that is within 4 cm of A and closer to B than to C.” Two conditions, so we want the overlap.

Condition 1 — within 4 cm of A. “Within” means a filled region: the disc inside the circle of radius 4 centred on A.
Condition 2 — closer to B than to C. The boundary between “closer to B” and “closer to C” is the perpendicular bisector of BC. The half-plane on B’s side is what we want.
Overlap. Shade only where the two overlap: the slice of the disc lying on B’s side of the bisector. Then test one point in your shading — is it really inside the circle and nearer B? If yes, you have shaded the right region.

Worked example 3 — the goat on a rope

“A goat is tied by a 6 m rope to a peg in the middle of a big field. What ground can it reach?”

Every reachable point is at most 6 m from the peg — the rope can be slack, so it is “within”, not “exactly”.
“Within 6 m of a point” is a filled disc of radius 6: a circle of grass the goat can graze, area \pi r^2 = 36\pi \approx 113\ \text{m}^2.
Now tie the goat instead to the corner of a barn wall: the wall blocks part of the circle, so the reachable region becomes part-circle plus, as the rope wraps the corner, a smaller arc — the same locus idea, just clipped by an obstacle.

The very same picture answers “how far does a phone mast reach?”. A mast with a 6 km range covers the disc of radius 6 km around it; where two masts’ discs overlap you have signal from both.

These are the mistakes that quietly cost marks in locus questions:

“Exactly” is a line; “within/less than” is a region. Read the wording. Exactly 5 cm from A is the circle (a curve); within 5 cm of A is the whole disc. Drawing a curve when they wanted a shaded area (or the reverse) is the most common slip.
Is the boundary included or not? “Less than 5 cm” excludes the rim (dashed circle); “5 cm or less” / “no more than” includes it (solid). Match the line style to whether the edge counts.
Combining conditions means the OVERLAP, not the union. “A and B” is where both hold — the intersection of the two regions, which is smaller. Students who shade everything covered by either rule have shaded far too much.
Always test a point. After shading, pick one point inside your region and check it obeys every condition, and pick one just outside and check it fails at least one. This catches a wrong side of a bisector instantly.

Loci are hiding inside the map app in your pocket. Your phone can estimate its distance to several things whose positions are known — GPS satellites overhead, or nearby phone masts. Each distance draws a locus: “I am this far from that mast” is a circle around the mast. One circle isn’t enough — you could be anywhere on it. But the circles from several masts cross at just one spot, and that overlap is you. It is exactly the two-condition region idea, run with three or four conditions at once. That trick even has a name: trilateration.

The same equidistant-from-two-points locus shows up in the real world in a lovely way: borders are sometimes drawn down the middle of a river so each country is nearer its own bank — that middle line is a perpendicular bisector. And when a referee wants the halfway line, they want the locus of points equally far from both goals. Distance rules, everywhere.