Geometry · GCSE Maths
Loci
GCSE Maths loci: equidistant from a point (circle), equidistant from a line (parallel line), equidistant from two points (perpendicular bisector), and from two lines (angle bisector).
Locus means a set of points obeying a rule. Fixed distance from a point: circle. Equal distance from two points: perpendicular bisector. Equal from two lines: angle bisector.
The important bits
What you need to know
- 1
The locus of points a fixed distance r from a point P is a circle, centre P, radius r.
- 2
The locus of points equidistant from two fixed points A and B is the perpendicular bisector of AB — a straight line through the midpoint of AB at 90°.
- 3
The locus of points equidistant from two fixed lines is the angle bisector(s) of the angle between the lines — two lines if the given lines cross.
- 4
The locus of points a fixed distance d from a line is a pair of parallel lines on either side, distance d from the line.
- 5
Construct with ruler and compasses on a diagram: circle for distance from a point; arcs to find the perpendicular bisector; arcs from a point on each line to find the angle bisector.
- 6
Regions combine loci with shading: “inside 4 cm of P and closer to AB than to CD” means inside the circle and on one side of the bisector.
- 7
On a coordinate grid, distance from (0, 0) of 5 gives x² + y² = 25. Distance from the x-axis of 3 gives y = 3 and y = −3.
- 8
Always draw the locus on the diagram the question gives. A freehand sketch without construction arcs may miss construction marks.
Quotations worth analysing
Short evidence. Real method.
“Equidistant from A and B → perpendicular bisector of AB”
Every point on the bisector is the same distance from A and from B. It is a line, not a circle, unless A and B coincide.
“Fixed distance r from P → circle, centre P, radius r”
“Within 3 cm of P” shades the interior of the circle. “More than 3 cm from P” is outside. “Exactly 3 cm” is the circle itself.
“Equidistant from two lines → angle bisector”
Two crossing lines give two bisectors — one for each pair of opposite regions. Parallel lines give a single midline parallel to both.
Go deeper
Constructing the perpendicular bisector
Join A and B. Open compasses more than half AB. Arc from A on both sides of AB. Same radius, arc from B so the arcs cross twice. Join the two crossing points: that line is the perpendicular bisector. Every point on it is equidistant from A and B. To find a point that is 5 cm from P and equidistant from A and B, draw the circle centre P radius 5 and the bisector; intersections are the candidates. If the question asks for a region, shade the side of the bisector closer to the point or line named in the question. Construction arcs must be visible for full marks on “show your construction” items.
Go deeper
Angle bisector and distance from a line
For two lines meeting at O, mark a point on each line equally far from O (same compass width). From those points, draw equal arcs into the angle. Join O to where those arcs cross: that is the angle bisector. Points on it are equidistant from both lines. For parallel lines, equidistant points form a line parallel to both, midway between them — not an angle bisector. Fixed distance d from a straight line L means two lines parallel to L, one d above and one d below in the diagram’s sense. A river bank problem might treat the bank as the line and the locus as the water’s edge on one side only.
Go deeper
Combining loci and regions
“A goat is tethered to a post P with a 8 m rope, and must stay on the near side of a wall through A and B.” Draw the circle radius 8 m about P and the perpendicular bisector of AB (or the half-plane on the correct side). The valid region is the overlap. “Closer to AB than to CD” means shade the side of the bisector of the angle between those lines, or the bisector of a segment joining a point on each line if they are parallel. List the loci, draw each, then shade the intersection. Coordinate loci: inside the circle x² + y² < 25; within 2 units of the line y = 1 means 1 − 2 < y < 1 + 2 if “within” means distance at most 2. Read whether the boundary is included.
See the idea in action
A point P is 5 cm from A and 5 cm from B. Describe the locus and construct it. Step 1: Points equidistant from A and B lie on the perpendicular bisector of AB. Step 2: Construct AB. Arc from A with radius more than ½AB, repeat from B. Step 3: Join the arc intersections: perpendicular bisector. Step 4: P lies on that line (any point on the bisector is equidistant from A and B). If also “3 cm from line L”: add two parallels 3 cm from L; P is where the bisector meets the correct parallel.
Exam technique
Turn knowledge into marks
Label each locus on the diagram (circle, bisector, parallel). Leave construction arcs visible when the question says “construct” or “show all construction lines”.
Common mistakes
Do not give these marks away
- 01
Drawing the angle bisector when the question asked for equidistance from two points (perpendicular bisector).
- 02
Shading the wrong side of a bisector when the question says “closer to A than to B”.
- 03
Using a freehand circle when construction with compasses was required.
The locus of points equidistant from two fixed points A and B is
AA circle with centre the midpoint of AB
BThe perpendicular bisector of AB
CThe angle bisector at A
DA line parallel to AB
Show the answer
The perpendicular bisector of AB. Equal distance from A and B defines the perpendicular bisector. A circle would be fixed distance from one point. Angle bisector is for two lines.
Quick questions
If this is the bit you searched
What is a locus?
The set of all points that satisfy a geometric condition, such as a fixed distance from a point or equal distance from two points.
How is locus different from distance from a line?
Distance from a line gives parallel lines on each side. Equidistant from two points gives one perpendicular bisector line.
Do I need compasses?
When the question says construct or asks for construction arcs, use compasses and ruler. Sketch-only may lose construction marks.
What if the two lines are parallel?
Equidistant from two parallel lines is a single line parallel to both, midway between them — not an angle bisector.