Locus (Higher Tier)

Locus (Higher Tier)

Locus

• A locus refers to the path a point follows according to a certain rule or condition.

• The plural of locus is loci.

Underlying Principles

• Any point equidistant from two other points will lie on the perpendicular bisector of the line joining those two points.

• Likewise, any point equidistant from the ends of a segment lies on the line segment’s perpendicular bisector.

• Any point on the angle bisector is equidistant from the sides of the angle.

Types of Locus

• The locus of points equidistant from a single point is a circle.

• The locus of points equidistant from two points is the perpendicular bisector of the line segment joining the two points.

• The locus of points equidistant from two intersecting lines is the angle bisector.

• The locus of points equidistant from two parallel lines is a line equidistant from and parallel to the two lines.

Locus in Three Dimensions

• In three dimensions, the locus of points equidistant from a single point is a sphere.

• The locus of points equidistant from two points is a plane whose normal is the line segment joining the two points.

Construction and Locus

• Construction is a method employed in geometry to draw shapes, angles or lines accurately using only a pair of compasses, a ruler and a pencil.

• A common GCSE style question might involve constructing loci under certain conditions for instance, all points closer to the line AB than the line CD.

• For construction of loci, key steps include using a compass to draw arcs from given points and using a rule to add in straight lines.

Bear in mind that locus problems often involve aspects of other parts of the syllabus, especially Pythagoras’ theorem. Make sure to revise these areas as well.