Construction: Loci

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Tutor: Bilal

Summary

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

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

A fixed distance from a given line

Two semicircles (one around either end of the line) and straight lines connecting the semicircles

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

Equidistant from two given lines

The angle bisector of the two lines

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

Equidistant from two given points

The perpendicular bisector of the line that connects the two points

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


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 than 3m3m from the point AA and is closer to the point AA than point BB.​


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 3m3m".

"Less than 3m3m" refers to a circle with radius 3m3m.


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 AA than point BB".

"Closer to point AA than point BB" refers to the perpendicular bisector of the line ABAB.


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





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Exercises

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FAQs - Frequently Asked Questions

What are loci?

How many loci are there?

How do you solve contextual loci problems?

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