A locus is the complete set of all points that satisfy a given condition. At GCSE, you draw loci accurately using a ruler and compasses, then identify regions that satisfy two or more conditions at once by shading the correct intersection. Freehand sketches do not earn marks.
What are the four standard loci you must know?
| Condition | Locus shape |
|---|---|
| All points equidistant from a fixed point P | Circle centred at P |
| All points equidistant from two fixed points A and B | Perpendicular bisector of AB |
| All points equidistant from two intersecting lines | Angle bisector of the angle between the lines |
| All points at a fixed distance from a line segment | Pair of parallel lines + semicircles at each end |
Memorising these four will let you set up almost any GCSE loci problem before picking up your compasses.
How do you construct the perpendicular bisector of a line segment?
The perpendicular bisector of AB is the locus of all points equidistant from A and from B. It passes through the midpoint of AB at 90°.
Construction steps:
- Open your compasses to more than half the length of AB.
- With the point on A, draw arcs above and below AB.
- Without adjusting the compass width, place the point on B and draw arcs above and below AB, crossing the first set.
- Join the two crossing points with a ruler and pencil.
The resulting line bisects AB at 90° and is the required locus.
Do not adjust the compass width between steps 2 and 3 — the two arcs must have the same radius for the construction to work.
How do you construct the bisector of an angle?
The angle bisector is the locus of all points equidistant from the two lines forming the angle. It divides the angle exactly in half.
Construction steps:
- Place the compass point at the angle's vertex V. Draw an arc that crosses both arms of the angle — label the crossings A and B.
- Place the compass point at A and draw an arc inside the angle.
- Without adjusting, place the point at B and draw an arc crossing the one from A — label the crossing C.
- Join V to C with a ruler.
VC is the angle bisector.
How do you construct a perpendicular from a point to a line?
This construction is needed when you must find the shortest distance from a point P to a line, or drop a perpendicular from P to that line.
- Open compasses from P wide enough to cross the line twice. Draw an arc — label the crossings A and B.
- With compasses from A (same or larger width), draw an arc on the far side of the line.
- Same width from B — arc crosses the first. Label the crossing Q.
- Join P to Q. The line PQ is perpendicular to the original line.
How do you solve a region problem with two conditions?
GCSE region problems say something like: "Shade the region within 4 cm of point A and closer to B than to C."
Worked example: On a scale drawing, towns A, B and C form a triangle. A mobile phone mast must be placed:
- Within 5 km of A (on the scale 1 cm : 1 km, this means within 5 cm of A on the diagram).
- Closer to B than to C.
Step 1: Draw a circle of radius 5 cm centred at A. The required region must lie inside this circle.
Step 2: Construct the perpendicular bisector of BC. Points closer to B than to C lie on the B-side of this bisector.
Step 3: Shade the region that is both inside the circle and on the B-side of the bisector. This is the intersection of the two conditions.
What does "closer to line PQ than to line RS" require?
This condition requires the angle bisector between lines PQ and RS. Points on one side of the angle bisector are closer to one line; points on the other side are closer to the other. Shade the region on the side corresponding to the stated condition.
For perpendicular lines, the angle bisector is at 45° — points within 45° of PQ are closer to PQ than to RS.
What mistakes should you avoid?
- Using a freehand arc. Every arc must be drawn with a compass. A freehand curved line is not a locus — it will not score marks.
- Adjusting the compass radius mid-construction. If the radius changes between the two arc pairs in a perpendicular bisector, the crossing point will be off-centre.
- Shading the wrong region. After identifying the boundary (bisector or circle), test a single point to confirm which side satisfies the condition before shading.
- Forgetting to leave construction marks visible. The examiner needs to see your arc marks to award method marks — do not erase them.
- Confusing the locus of a distance from a line with the locus of a distance from a point. Distance from a point gives a circle; distance from an infinite line gives two parallel lines; distance from a line segment gives two parallel lines with semicircular caps at each end.
Frequently asked questions
What does "equidistant from A and B" mean in everyday language?
It means the same distance from A as from B. If you are standing at any point on the perpendicular bisector of AB, your distance to A equals your distance to B — exactly. This is why the perpendicular bisector is the correct locus. You can verify any point on it by measuring directly, or by checking that it lies on the bisector construction you drew.
When do I draw a solid line versus a dashed line for a locus boundary?
Use a solid line when the boundary is included in the region (the condition says "within 3 cm" or "at most 3 cm" — the boundary circle is included). Use a dashed line when the boundary is excluded ("less than 3 cm" — strictly inside the circle). Many GCSE mark schemes accept either, but being precise shows higher mathematical understanding.
Can a locus include multiple regions?
Yes. "All points equidistant from two parallel lines" gives a single line midway between them — one region. But "all points equidistant from the sides of a triangle" involves three angle bisectors meeting at the incentre, and the region of a complex multi-condition locus can be a polygon, a partial disc, or an arc. Always draw every boundary first, then identify the intersection.
Do I need a protractor for these constructions?
No — proper ruler-and-compasses constructions do not use a protractor. A 60° angle, a perpendicular, and a bisected angle are all constructed from arc intersections alone. Using a protractor and ruler instead of compasses will not earn full marks on a GCSE construction question, even if the angle appears correct.
For Socratic GCSE geometry support with Professor Pi, visit AI Tutors.