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:
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:
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 than3mfrom the pointAand is closer to the pointAthan pointB.
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.
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.