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.

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.

*S*represents the school and point

*H*represents a point 5 km away.

*H*is therefore a possible position of Rhiannon's house.

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

We say that the

**locus***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**

**locus***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**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.

Now try interpreting a locus diagram yourself.

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.

As you can see, the point traces out a

**straight line***y = ax + c*. It has length and position, but no breadth and is therefore one-dimensional.straight line

**parallel***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.

Look at the

*Geometric Construction*unit if you do not already know how to construct parallel lines.

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)**.

The two points where these loci

**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.

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***Geometric Construction*unit for more on such lines.)

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.

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***Geometric Construction*unit.

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.

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.

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.

Summary

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