Loci and Regions
The path of points that obey a rule
A locus (plural loci) is the set of all points
that satisfy a given condition. Instead of one answer, a rule about distance usually traces
out a whole shape — every point on it obeys the rule, and no point off it does.
-
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;
-
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.
Loci, and regions built from them
Each rule has a standard locus. Combine the rules and you get a region —
the part of the plane where all the conditions hold at once.
- fixed distance from a point → a circle;
- 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 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.