Loci

 

What Are Loci?

A locus (plural: loci) is a set of points that satisfy a specific condition or rule.

You might be asked to construct loci based on real-world descriptions, such as:

"A point 3 cm from a given point"

"The region closer to point A than point B"

These constructions often involve using a compass, ruler, and protractor.

 

Common Types of Loci

Locus Type	Description
Fixed distance from a point	Forms a circle with the point as the centre. Use compasses.
Fixed distance from a line	Forms a parallel band with semi-circular ends (like a running track).
Equidistant from two points	Forms the perpendicular bisector of the line between the points.
Equidistant from two intersecting lines	Forms the angle bisector between the two lines.

 

How to Find Regions That Satisfy Conditions

To identify the region that meets a condition:

• Draw the appropriate construction (circle, bisector, etc.)
• Tick the area(s) that satisfy the condition
• Cross the ones that don’t
• The final region is the one with only ticks (no crosses)

 

Examples of Region Conditions

Closer to point A than point B $\to$ Use perpendicular bisector of AB

Within 4 cm of point P $\to$ Draw a circle of radius 4 cm centered at P

Closer to line AB than AC $\to$ Use angle bisector at A

 

 

 

 

Worked Example 1: Closer to One Side of a Triangle

You are given triangle ABC and asked to shade the region closer to side AC than side BC.

Step-by-step:

1. Construct the angle bisector of angle A.
2. Use a compass to draw arcs from A that cut sides AB and AC.
3. With the compass at each of those intersection points, draw arcs that intersect each other.
4. Draw a straight line from A through this intersection – this is the angle bisector.
5. Shade the region between the bisector and side AC.

 

Worked Example 2: House & Radio Mast Problem

A house lies between Town A and Town B. Radio mast R provides signal if the house is:

• Closer to Town A than Town B, OR
• Outside a circle 5 miles from mast S.

Step-by-step:

1. Draw the line joining Town A to Town B.
2. Construct the perpendicular bisector – this shows points equidistant from both towns.
3. The house lies on the Town B side of the bisector $\to$ Condition NOT satisfied.
4. Now draw a circle of radius 5 cm around mast S (since 1 cm = 1 mile).
5. The house lies outside the circle, so this condition IS satisfied.

Final Conclusion: The house satisfies one condition, so it receives signal from mast R.