G2 - Loci and shortest distances

You already know

A compass draws points at one fixed distance from its centre. A perpendicular bisector passes halfway between two points, and an angle bisector splits an angle equally.

Locus

The set of all points which satisfy a stated condition. The plural is loci. For example, the locus of points exactly $3\,\mathrm{cm}$ from $A$ is the circumference of a circle with centre $A$ and radius $3\,\mathrm{cm}$. The interior is not part of that locus.

Worked example 1: one fixed distance

Question. Describe and construct the locus of points exactly $4\,\mathrm{cm}$ from point $A$.

Plan. Use $A$ as the centre and the stated distance as the radius.

1. Set the compasses to $4\,\mathrm{cm}$: this fixes the distance.
2. Place the point at $A$ and draw the complete circle: every direction must be included.
3. Leave the inside unshaded: its points are less than $4\,\mathrm{cm}$ from $A$.

Why this step? A complete circle includes every possible direction from $A$, not just positions left and right of it.

Answer. The circumference of a circle with centre $A$ and radius $4\,\mathrm{cm}$.

Check. Every boundary point is $4\,\mathrm{cm}$ from $A$; the centre itself is not.

Worked example 2: include the interior

Question. A point $P$ must be at most $5\,\mathrm{cm}$ from $B$. Describe the required region.

Plan. Construct the limiting circle, then include the positions nearer $B$.

1. Draw the circle with centre $B$ and radius $5\,\mathrm{cm}$: its circumference is the boundary.
2. Shade its interior and include the circumference: both shorter distances and exactly $5\,\mathrm{cm}$ work.

Why this step? “At most” means “less than or equal to”, so the boundary belongs to the answer.

Answer. The closed circular region of radius $5\,\mathrm{cm}$ centred at $B$.

Check. Distances $0\,\mathrm{cm}$, $4\,\mathrm{cm}$ and $5\,\mathrm{cm}$ work; $6\,\mathrm{cm}$ does not.

Watch out

Wrong: shade a circle's interior when the question says “exactly”. Right: exactly one distance means its circumference alone.

Equidistant from two points

For distinct points $A$ and $B$,

exactly when $P$ lies on the perpendicular bisector of $AB$.
Not on the formulae sheet: learn it.

Worked example 3: construct a shared boundary

Question. Points $A$ and $B$ are $8\,\mathrm{cm}$ apart. Construct the locus of points equally far from them.

Plan. Construct the perpendicular bisector of $AB$.

1. Use a compass opening greater than $4\,\mathrm{cm}$: the equal arcs must cross.
2. From $A$ and $B$, draw equal arcs above and below $AB$: label their crossings.
3. Join the crossings and extend the line: this is the complete locus.

Why this step? Equal distances do not mean a fixed distance. Points further along the bisector are further from both endpoints, but the two distances stay equal.

Answer. The perpendicular bisector of $AB$, crossing it $4\,\mathrm{cm}$ from each end.

Check. The midpoint is on the locus. Neither $A$ nor $B$ is, because their distances to themselves are zero.

Worked example 4: choose one side

Question. A location must be closer to $D$ than to $C$. Describe the region when $CD=12\,\mathrm{cm}$.

Plan. Draw the equal-distance boundary, then choose the side containing $D$.

1. Construct the perpendicular bisector of $CD$: it crosses $CD$ at $6\,\mathrm{cm}$.
2. Shade the half-plane containing $D$: moving to that side makes $D$ nearer.
3. Exclude the bisector: points on it are equally far away, rather than closer to $D$.

Why this step? Testing $D$ itself settles which side to choose: its distance from $D$ is zero.

Answer. The side of the perpendicular bisector containing $D$, with the boundary excluded.

Check. $D$ is included; $C$ and the midpoint are excluded.

Watch out

Wrong: draw a circle with centre the midpoint for equal distances from two points. Right: use the perpendicular bisector. A circle would unnecessarily fix the distance from its centre.