Loci
Introduction

The word 'locus' means 'place' in Latin. So when we talk about finding the locus of an object, we want to find the places where it might be. For instance, looking at the map below, we are told that a certain object lies south of the river and west of the forest. We can work out that it must lie in the shaded area.



The loci we deal with here are more precise than this, but they are found by narrowing down possible positions in much the same way.
Loci and points
If we are told that a pupil, Rhiannon, lives 5 km from her school, but we do not know in what direction, how can we show the possible locations of her home? In Fig.1 below
point
  • A point has no properties except position. It is an object with zero dimensions.
  • Points in the x-y plane can be specified using x and y coordinates.
point
S represents the school and point H represents a point 5 km away. H is therefore a possible position of Rhiannon's house.

Figure 1.   Rhiannon's home and school.
Display as fullscreen


The school is definitely located at the given position. Move point H to see all the places where Rhiannon's house might be.

What is the shape that point H traces out?
  • Click here to mark the question


We say that the
locus
A locus is a set of points that all satisfy a particular condition. For instance, the two-dimensional locus of points equidistant from two points A and B is the perpendicular bisector of the line segment AB.
locus
of all points 5 km from the school is a circle (with a
radius
A straight line segment joining the centre of a circle with a point on its circumference (or the length of this line).
radius
of 5 km). If we were asked to draw the
locus
A locus is a set of points that all satisfy a particular condition. For instance, the two-dimensional locus of points equidistant from two points A and B is the perpendicular bisector of the line segment AB.
locus
representing possible positions of Rhiannon's house (labelling it H), we would draw the circle shown in Fig.1.

We are now told that the school only takes children who live not more than 6 km away. We represent this catchment
area
Area is a measure of surface size and is calculated in square units (e.g. cm, m, sq.ft.).
area
in the following way.

Figure 2.   The locus of all points not more than 6 km from the school.
Display as fullscreen


All points on the circle are 6 km from the school and so are part of the catchment area. Also, all points within the circle are less than 6 km from the school and so also form part of the catchment area. Hence we shade in the circle to represent such a locus.

A dotted, rather than a solid, line is used when the points directly on the boundary are not included in the locus. For instance, look at Fig.3. It shows the locus of points outside the catchment area of the school (more than 6 km from the school). Points on the dotted line are 6 km from the school and so are part of the catchment area, hence they are not part of the locus of points outside the catchment area. The circular boundary line is dotted to show this.

Figure 3.   The locus of points outside the catchment area.
Display as fullscreen


Now try interpreting a locus diagram yourself.

Figure 4.   A locus.
Display as fullscreen
The above locus represents the possible positions of the home of a second pupil, Paul. Which is the correct description of this locus?
  • No. This locus would include all points 5km or less too.
  • No. This locus would include points that are 5km away, but the diagram shows a dotted line at these positions
  • Click here to mark the question
Loci and lines
When we talk about 'distance from a line', we always mean the perpendicular distance from the line. Drag the point, P, in Fig.5 below to see it move keeping a distance of 2 cm from line 1.


Click on the figure below to interact with the model.

 Figure 5.  A point 2 cm from a line.



As you can see, the point traces out a
straight line
A straight line is a set of points related by an equation of the form y = ax + c. It has length and position, but no breadth and is therefore one-dimensional.
straight line
parallel
Two lines, curves or planes are said to be parallel if the perpendicular distance between them is always the same.
parallel
to line 1. This is not the locus of all points 2 cm away from line 1, however. Such points also lie on the other side of line 1. Hence the loci, L, of all points 2 cm from line 1 is as shown in Fig.6 below.

Figure 6.   The locus of all points 2 cm from line 1.
Display as fullscreen


Look at the Geometric Construction unit if you do not already know how to construct parallel lines.

Which of the following diagrams represents the locus of all points less than 1 cm from the
line segment
A line segment is the set of points on the straight line between any two points, including the two endpoints themselves.
line segment
AB?
  • Well done! This locus is constructed by drawing two parallel lines and using a compass to create the 1 cm radius semicircles centred at A and B.
  • Click here to mark the question
Look at the lines in Fig.7 below. They are all parallel. Which of the following statements is true?
  • Click here to mark the question
Figure 7.  
Display as fullscreen
Equidistance from two points
Look at the model below. Imagine you live 2 km from your school and 2 km from the museum. To find where you live, we would draw the locus, A, of all points 2 km from your school (2) and then the locus, B, of all the points 2 km from the museum (3).

Figure 8.   The school and the museum (and associated loci).
Display as fullscreen


The two points where these loci
intersect
To intersect is to have a common point or points. For example, two lines intersect at a point and two planes intersect at a straight line. The point at which two or more lines intersect is called a vertex.
intersect
(4) are the points where you might live since they are the only points that are both 2 km from the school and 2 km from the museum.

But what if, instead of knowing that you lived 2 km away from both the school and the museum, you knew only that you were the same distance from each? The radius of each of these circular loci could be any value. Adjust the radii by dragging the control point of the circles in Fig.9 below.

Figure 9.   Loci centred on the school and the museum.
Display as fullscreen


The points where the circular loci intersect again indicate points of equidistance. Whatever the radius is that is shared by the loci, the points of equidistance from the school and the museum all lie along a straight line. This line is also called the
perpendicular bisector
A perpendicular bisector is a line that cuts in half a given line segment and forms a 90° angle with it.
perpendicular bisector
of the line between the school and museum. (Once again, see the Geometric Construction unit for more on such lines.)

Look at the four lines in Fig.10 below. Which of the lines represents the locus of points
equidistant
Items that are at an equal distance from an identified point, line or plane are said to be equidistant from it.
equidistant
from P and Q?
  • Click here to mark the question
Figure 10.   Points P and Q.
Display as fullscreen
The map in Fig.11 below shows an island. On it are marked 3 towns: Ashville (A), Barnton (B), and Critchley (C). There is also a fourth town, Dreyhorn (D), on the island. Which of the following describes the position of Dreyhorn as shown by the loci drawn in Fig.11?
  • Click here to mark the question
Figure 11.   Map of an island and its town.
Display as fullscreen
Equidistance from intersecting lines
Below, in Fig.12 are two intersecting lines. The point X can be moved, but it is always equidistant from lines 1 and 2.


Click on the figure below to interact with the model.

 Figure 12.  Intersecting lines.



We can see from the model that X is limited to movement along a straight line. In fact, this straight line is the locus of all points equidistant from lines 1 and 2. You may notice that this straight line locus has another important property: it bisects the
angle
An angle is a measure of turning. Angles are measured in degrees. The symbol for an angle is .
angle
between the two lines. You can see how to draw such lines in the Geometric Construction unit.

Look at the pattern in Fig.13 below. Which lines represent the locus of all points equidistant from lines c and g?
  • Click here to mark the question
Figure 13.   Intersecting lines.
Display as fullscreen


So the black lines are, together, the locus of points equidistant from lines c and g. (If this is not immediately obvious, imagine the intesecting lines c and g as four sets of the 'V' shape shown in Fig.12.)
Compound loci
We call groups of loci compound loci. Often we have to locate a region using compound loci produced by a set of information. We have already seen some simple examples of compound loci, but here we may have to use any combination of the types of loci we have already come across to locate a particular region.

The diagram below shows a coastal region. A small boat is stranded in the sea somewhere in the waters off the coast. The captain radios for help, saying that they seem to be closer to the south shore than the north shore. The strength of the radio signal indicates that they are not more than 1 km from Sea Rescue headquarters (point H). We must locate the region in which the search should begin.

Figure 14.   Coastal map.
Display as fullscreen


First, we need to extract the key information from the paragraph above. Three separate pieces of information have been given for the position of the boat.

  • The boat is in the sea.

  • The boat is closer to the south shore than the north.

  • The boat is 1 km or less from point H.



Wherever the boat is, it must satisfy all these conditions.

Figure 15.   Constructing the answer.
Display as fullscreen


We start by finding the locus of all points equidistant from the north and south shores (2). This line is dotted since the points on it are not closer to the south shore than the north: they are the same distance from both. All points closer to the south shore are underneath the line, so we shade this region (3). The shaded area now represents points in the sea that are closer to the south shore than the north.

We now look at the third locus. All points 1 km from H form a circle (radius 1 km) with H at the centre (4). All points less than 1 km from H lie within the circle, so we shade this region too (5).

We know that the boat is in the sea within the region that has been shaded twice. We can label this region R (6). This is the area in which the rescue services should look for the boat.

In the diagram below, towns A and B are marked. Region R is the locus of …
  • Click here to mark the question
Figure 16.   Region R.
Display as fullscreen
Summary

The locus of all points at a distance d from a fixed point, O, is represented by a circle centre O, radius d.
The locus of all points at a distance d from a fixed line is represented by two parallel lines either side of it at a distance d away.
The locus of all points at a distance d from a fixed line segment is represented by two parallel lines either side of it at a distance d away, plus two semicircles of radius d centred on the end points of the line segment.
The locus of all points equidistant from two points, A and B, is the perpendicular bisector of the line AB.
The locus of all points equidistant from two intersecting lines is made up of the two lines that bisect the angles formed where the original lines intersect.

Exercises
Figure 17.   Four loci.
Display as fullscreen
1. Match each of the red loci in Fig.17 to its description below.
  • Locus 1
    Locus 2
    Locus 3
    Locus 4
  • Click here to mark the question
Figure 18.  
Display as fullscreen
2. Use the diagram in Fig.18 above to complete the sentence below.

  • The locus of points equidistant from the two lines is the straight line through and .
  • Click here to mark the question
Figure 19.  
Display as fullscreen
3. A ship is located at least 300 m from the coast, c, but less than 200 m from the lighthouse, L. The region in which the ship can be found has been shaded. Which of the diagrams in Fig.19 above represents this region?
  • Click here to mark the question
Figure 20.  
Display as fullscreen
4. The diving pool in Fig.20 above is to have an area roped off around it since anyone less than 5 m from the pool is in danger of being soaked. Which of the diagrams represents the danger area (in grey)?
  • Click here to mark the question


Figure 21.   A map of Rome.
Display as fullscreen


5. Fig.21 above is a map of part of Rome. From some old records, archeologists are trying to find the site of an ancient temple that is now buried beneath the city. They have already located 12 possible sites, but they have more information yet. They also know that the temple was less than 2.5 km from both the Colosseum and the Castel Sant'Angelo, and over 2.25 km from the Basilica of St. Peter's. Which of the 12 sites already located and shown above fit this description? (Hint: there is more than one.)
  • Click here to mark the question
Well done!
Try again!