What are loci and constructions?
A locus (loci is plural) is a line/shape/path that is determined by following a rule – eg always being 2m away from a wall. You may be asked to construct a locus, although the language used in exam questions don’t always mention these words as questions are often based on real world situations.
It is an excuse to get the maths toys out – rulers and compasses in particular!
You will often be working with scale drawings with this topic.
There are some basic loci – like a circle for being a certain distance away from a point (eg distance a mobile phone mast can reach) and a straight line for being a certain distance away from another line (like the example above – always staying 2m away from a wall).
These notes focus on three of the more complicated constructions and their uses:
- Perpendicular bisector
Maths: A line that cuts another one exactly in half (bisect) but also crosses it at a right angle (perpendicular)
Uses: Shows a path that is equidistant (equal distance) between two places, if you are on one side or the other of this line you know you are closer to one place than the other.
2. Perpendicular to a line, from a point
Maths: As per the title really! This is drawing a line, perpendicular to another, but the line has to go through a particular point
Uses: Shows the shortest distance from the point to a line – eg the shortest distance from a lighthouse to the coastline
3. Angle bisector
Maths: A line that cuts an angle exactly in half (bisect), either side of the line shows you are closer to one line making the angle than the other
Uses: Shows a region on a map/diagram that is closer to one side than another
- On the triangle below indicate the region that is closer to the side AC than AB.
3 – This is an angle bisector – but notice how nothing about angles gets mentioned in the question!
To start we open our compasses at comfortable distance (not too far, not too short) and we draw arcs from sides AC and AB with the compasses point on A
Leaving the compasses open at the same distance (this is why you don’t want a loose and wobbly set!) we now draw arcs from those first two arcs
(Note: It was luck, not judgement or anything that meant the arcs crossed on or very close to the side BC, this is neither expected nor required)
The line going through A and where the arcs cross is the angle bisector
Your final answer will be on the side of this line that AC is on
2. A house lies between two towns A and B as shown on the scale diagram below.
Two masts, located at points R and S, provide the area on the map with radio signals.
The house will receive its radio signal from the mast at point R if it is:
(a) closer to town A than B, OR
(b) outside a region five miles from the mast at point S.
If neither it will receive its mobile phone signal from the mast at point S.
Showing your working carefully on the scale diagram below, determine whether the house receives radio signals from the mast at point R or the mast at point S.
1 – To find (a) we need the perpendicular bisector of AB – notice there is no line drawn between A and B to start off with, but we can easily add one
Open compasses more than half the distance (no precise distance) and draw arcs both above and below the line joining A and B – do this from both point A and point B
The perpendicular bisector is the line that passes through both points where each set of arcs meet
The house is closer to town B
We know need to check (b) – notice the question says “or” (rather than “and”) so only one of the conditions needs to be satisfied for the house to get its TV signal from the mast at R
This is a basic locus – it is an arc (part of a circle) drawn – to scale – with the centre at S and compasses open at 5 cm
You can now see that the house is outside the region five miles from the mast at point S and so you can answer the question
Although the house is closer to town B than town A it is outside a region five miles from mast S, so the house receives its radio signal from the mast at point R.
(Note in this case if you had checked (b) first you would not have to check (a) as well – this is fine but your final answer should explain why you only needed to check one of them.)