Loci
Geometry · Loci
Syllabus tag: KCSE | Mathematics | Form 4 | Topic 7 Loci
Lesson objectives
By the end of this topic, you should be able to:
- Define a locus and describe common loci.
- Construct the loci involving inequalities, chords and points under given conditions.
- Construct the locus of points subtending a constant angle.
- Apply loci to solve problems in two dimensions.
Loci
A locus is the set of all points satisfying a given condition. The plural is loci.
a) The four common loci
| Condition | The locus |
|---|---|
| A fixed distance from a point | a circle of that radius |
| Equidistant from two points | the perpendicular bisector of the join |
| Equidistant from two lines | the angle bisector |
| A fixed distance from a line | two parallel lines, one each side |
b) Equidistant from two points
The locus is the perpendicular bisector of the line joining them.
Every point on it is the same distance from both. That is exactly the defining property of the bisector.
c) Equidistant from two lines
The locus is the angle bisector of the angle between them.
If the lines cross, there are two bisectors, perpendicular to each other. Give both unless the question restricts the region.
d) The locus of a constant angle
Let a fixed segment AB subtend a constant angle. The locus is an arc on each side of AB.
This follows from the circle theorem that angles in the same segment are equal.
When the angle is 90°, the locus is a circle with AB as diameter. The angle in a semicircle is a right angle.
e) Loci involving inequalities
An inequality gives a region, not a line.
"Less than 3 cm from P" is the inside of a circle of radius 3 centred at P.
"Nearer to A than to B" is the side of the perpendicular bisector containing A.
Draw the boundary first, decide which side satisfies the condition, then shade.
Use a dashed boundary when it is excluded, and a solid one when included. That is the same convention as for inequalities on a graph.
f) Combining conditions
Most questions impose two or three conditions at once.
Construct each locus separately, in pencil, leaving all construction arcs visible.
The answer is where all the conditions are satisfied at once. That may be a region, a line, or just one or two points.
Mark the final answer clearly and describe it in words as well. An unlabelled diagram loses marks that a short description would earn.
g) Practical advice
Work with a sharp pencil and compasses, and do not erase construction arcs.
Marks are given for the construction as much as the answer. An accurate answer with no visible method scores poorly.
Read whether the question wants a locus of points, a region, or a specific point.
h) Where this is used
Positioning a facility equidistant from several towns. Safety zones round a hazard. Boundary disputes over land. Any design constrained by distance conditions.
Words to know
- Locus -- the set of all points satisfying a given condition.
- Perpendicular bisector -- the locus of points equidistant from two fixed points.
- Angle bisector -- the locus of points equidistant from two intersecting lines.
- Constant angle locus -- the arc from which a segment subtends a fixed angle.
- Intersecting loci -- the points satisfying two or more conditions at once.
:::checkpoint Check yourself
- What is the locus of points 4 cm from a fixed point?
- What is the locus of points equidistant from two fixed points?
- What is the locus of points where AB subtends 90°?
- How do you show a boundary that is excluded from a region? :::
Bridge to practice
The exercises begin with the four basic loci and their descriptions, move through inequalities and the constant angle locus, and finish with intersecting loci and applications. After drawing each locus, check whether a matching second branch exists on the other side.