Loci and Regions
Grade 5What is a locus?
A locus (plural: loci) is the set of all points that satisfy a given condition. Think of it as the path traced by a point that follows a rule.
The four standard loci at GCSE: (1) a fixed distance from a point → a circle; (2) a fixed distance from a line → two parallel lines and semicircles at ends; (3) equidistant from two points → perpendicular bisector; (4) equidistant from two lines → angle bisector.
Combining loci into regions
Many questions ask you to shade a region — the intersection of two or more loci conditions. Draw each boundary first, then shade where both conditions are met.
Read "within 3 cm of point A" as: shade the inside of the circle of radius 3 cm centred on A. Read "closer to line AB than line BC" as: shade the side of the angle bisector nearer AB.
Worked example
EXAMPLE
On a scale diagram, shade the region that is less than 4 cm from point P and closer to A than to B.
Draw a circle of radius 4 cm centred on P — points inside satisfy the first condition. Construct the perpendicular bisector of AB — points on A's side satisfy the second. The required region is inside the circle and on the A-side of the perpendicular bisector (the intersection of the two regions).
