📍 Getting in Line: Mastering Simple Loci!

Ever wondered how engineers decide where to put street-lights, or how gardeners space out sprinklers? The secret lies in **loci**—the set of points that satisfy certain conditions. Understanding loci helps O-Level students tackle construction questions accurately and build strong geometric intuition for real-world tasks.

What Is a Locus?

A locus (plural *loci*) is **all points** that meet a given rule.
Typical rules at Core level are: fixed distance from a point (circle), fixed distance from a line (parallel lines), or equidistant from two points (perpendicular bisector).
When drawing, keep compass radius constant and use sharp, light arcs for accuracy.

Worked Example: Placing a Lamp Post

Problem: A straight road

AB

80\,\text{m}

long. A garden path meets

AB

B

forming an angle of

90^{\circ}

. Construct and shade the region where a lamp post can be placed such that it is (i) no more than

20\,\text{m}

from

A

, and (ii) nearer to the garden path than to the road.

Solution (step-by-step):
1. Draw road

AB

to scale and the perpendicular garden path at

B

.
2. Condition (i): With centre

A

and radius

20\,\text{m}

, draw an arc—this is the circle

x^{2}+y^{2}=20^{2}

in coordinate terms.
3. Condition (ii): Being "nearer to the path than to the road" means constructing the angle bisector between the road and the path. Use your compass to mark equal arcs on both lines, then join their intersection to

B

to get the bisector.
4. The required locus is the part of the circle **inside** the half-plane defined by the bisector. Shade this region lightly.
5. Double-check with your ruler and compass: every shaded point should measure

\le 20\,\text{m}

from

A

and be closer to the path.