Loci

A locus is the set of all points that satisfy a given condition. Loci connect geometric constructions with equations and constraints.

Basic loci patterns

• Fixed distance \(r\) from point \(A\): a circle center \(A\), radius \(r\).
• Fixed distance from line \(l\): two parallel lines on each side of \(l\).
• Equidistant from points \(A,B\): perpendicular bisector of \(AB\).
• Equidistant from two intersecting lines: angle bisectors.

Constructing loci

1. Translate condition into known locus type.
2. Construct each required locus accurately.
3. Find intersection of loci for final solution points/region.

Loci and inequalities

Conditions like “distance at most 3 cm from line” describe a region, not a single line.

Boundary inclusion matters:

• \(\le\): boundary included (solid)
• \(<\): boundary excluded

Algebra links

Coordinate forms:

• Circle: \((x-h)^2+(y-k)^2=r^2\)
• Perpendicular bisector from equal-distance condition \(PA=PB\)

Modeling with loci

Loci are used for coverage regions, safe-distance boundaries, and optimal placement constraints.

Exam strategy

• Underline keywords: equidistant, fixed distance, within, at least.
• Draw boundaries first, then shade valid region.
• State whether endpoints/boundaries are included.
• Use intersections of loci to justify final points.

History

Locus ideas emerged from classical compass-and-straightedge geometry and later became a core part of analytic geometry and optimization problems.

Checkpoints

Applications

• Cell/Wi-Fi planning: coverage zones around transmitters.
• Urban design: buffer distances from roads or facilities.
• Robotics: safe navigation boundaries and target zones.

References

• Standard high-school geometry content on loci and construction.
• Analytic geometry materials linking loci to equations.