Construction: Loci

In a nutshell

Loci is the plural of locus. A locus is a set of points that obey a certain rule. There are four main loci that you need to be able to perform using a ruler and compass. These four loci can be applied to solve problems in context.

The four loci

Here are the four loci that you need to know.

LOCI	DESCRIPTION	DIAGRAM
A fixed distance from a given point	A circle with the point as the centre
A fixed distance from a given line	Two semicircles (one around either end of the line) and straight lines connecting the semicircles
Equidistant from two given lines	The angle bisector of the two lines
Equidistant from two given points	The perpendicular bisector of the line that connects the two points

Constructing a fixed distance from a given point

procedure

1.	Use the compass and ruler to measure the given distance.
2.	Draw a circle of the given distance about the point.

Visually:

Maths; Angles and geometry; KS4 Year 10; Construction: Loci

Constructing a fixed distance from a given line

procedure

1.	Use the compass and ruler to measure the given distance.
2.	Draw circles of the given distance about each end of the line.
3.	Draw lines that are parallel to the given line that touch the circles.

Visually:

Maths; Angles and geometry; KS4 Year 10; Construction: Loci

Note: The procedures for constructing perpendicular and angle bisectors are covered in the lesson Constructions.

Loci in context

You may have to apply loci to more contextual problems. This may also involve identifying and shading regions.

Example

In the map below, one box represents one square metre. Shade in the region that is less than $3m$ from the point $A$ and is closer to the point $A$ than point $B$ .

Maths; Angles and geometry; KS4 Year 10; Construction: Loci

Deal with one locus at a time.

First, identify the locus that is associated with the phrase "less than $3m$ ".

"Less than $3m$ " refers to a circle with radius $3m$ .

Construct the relevant locus and mark dashes that indicate the region. In this case, the region is inside the circle.

Maths; Angles and geometry; KS4 Year 10; Construction: Loci

Do the same with the phrase "closer to point $A$ than point $B$ ".

"Closer to point $A$ than point $B$ " refers to the perpendicular bisector of the line $AB$ .

Maths; Angles and geometry; KS4 Year 10; Construction: Loci

Shade in the region that satisfies all the loci.

Maths; Angles and geometry; KS4 Year 10; Construction: Loci