A locus is the set of all points that follow one rule. In IGCSE Mathematics the rule is nearly always about distance: from a point, from a line, or from two points at once.
This lesson sits in constructions and loci and uses the constructions from perpendicular bisectors and angle bisectors.
Which rule gives which shape?
| Rule | Locus |
|---|---|
| A fixed distance from a point | Circle centred on the point |
| A fixed distance from a long straight line | Two parallel lines, one on each side |
| A fixed distance from a line segment | Two parallel segments joined by semicircles at the ends |
| Equidistant from two points | Perpendicular bisector of the segment between them |
| Equidistant from two lines that meet | Angle bisector of the angle between them |
Always decide first whether the rule says exactly or within. Exactly gives a boundary. Within gives a region.
Worked example
A goat is tied by a rope so that it can reach only points that are exactly 2 m from a straight fence rail PQ, where PQ = 6 m. Describe the locus and find its total length.
Step 1, sides of the rail: points 2 m from PQ on each side form two lines parallel to PQ, each 6 m long.
Step 2, ends of the rail: near P and near Q the points are 2 m from the end point, so they lie on semicircles of radius 2 m.
Step 3, find the length: the two straight parts total 6 + 6 = 12 m. The two semicircles make one full circle of radius 2 m, with circumference 2 × π × 2 = 4π m.
Step 4, add: 12 + 4π = 12 + 12.57 = 24.57, so the locus is about 24.6 m long.
Check: 4π is about 12.566, and 12 + 12.566 = 24.566. That rounds to 24.6. Both calculations agree.
If the question said within 2 m, the region would also include all the ground inside this boundary.
The mistake to watch for
A common slip is to draw a rectangle with square corners around the segment.
Mistaken answer: the locus is a rectangle 10 m by 4 m around the rail, with perimeter 28 m.
The corners of that rectangle are further than 2 m from the rail, so they break the rule.
The correction is to test a point: each corner is 2√2 ≈ 2.8 m from the nearest end of PQ, so it is not on the locus. Replace each pair of corners with a semicircle of radius 2 m. The perimeter then drops from 28 m to about 24.6 m.
Check yourself
Try each question, then open the answer.
1. Describe the locus of points 4 cm from a fixed point O, and find its length to 1 decimal place.
Show answer
A circle of radius 4 cm centred at O. The circumference is 2 × π × 4 = 8π ≈ 25.1 cm.
2. Describe the locus of points exactly 3 cm from an infinitely long straight line.
Show answer
Two straight lines, parallel to the given line, 3 cm away on each side.
3. Find the area of the region of points within 5 cm of a fixed point O, to 1 decimal place.
Show answer
The region is a circle of radius 5 cm. Area = π × 5² = 25π ≈ 78.5 cm².
Where this leads next
When two rules apply at once, you draw both loci and look for the overlap, as in combining two loci to locate feasible positions. The non-calculator working trainer helps with exact answers such as 6 + 6 + 4π.
Some students know the rule table but lose marks when the sentence is wordy. Our teachers practise that translation in online one-to-one Mathematics tuition.