Geometry & measures · Constructions & loci
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Loci & regions
A locus as the set of points obeying a rule, the four standard loci, and how to combine several conditions — with scale drawings — to shade the region that satisfies all of them at once.
Geometry & measures · Constructions & loci
Loci & regions
A locus as the set of points obeying a rule, the four standard loci, and how to combine several conditions — with scale drawings — to shade the region that satisfies all of them at once.
Why it works
A locus is simply "every point that obeys the rule". Ask *which points qualify?* and the shape draws itself. Four rules cover almost every GCSE question:| Rule | Locus |
|---|---|
| A fixed distance from a point | a circle, centre , radius |
| A fixed distance from a line segment | a racetrack: two parallel lines away, joined by semicircular ends |
| Equidistant from two points and | the perpendicular bisector of |
| Equidistant from two lines | the angle bisector between them |
"Nearer to than to " is a region, not a line. Draw the boundary first (the perpendicular bisector), then decide which side satisfies the rule and shade it. Same for "less than 4 cm from ": draw the circle, then shade inside. The boundary line is the tie — the points exactly equidistant.
Combining conditions: shade the overlap. Real questions stack two or three rules ("less than 5 m from the wall and nearer to hedge than hedge "). Draw every boundary, work out which side of each one is allowed, and shade only the region where all the conditions hold at once. A key with your shading, and the boundaries left drawn in, is what earns the marks.
Scale is where the arithmetic lives. A plan drawn at cm to m turns "within 30 m of the mast" into a circle of radius cm on the page. Convert the real distance to a drawing distance before setting the compasses, and multiply back the other way when reading a length off the plan. Getting the geometry right but the scale wrong is the most expensive slip in this topic.