Maths

Loci

A locus is a path. The path is formed by a point which moves according to some rule.

The plural of locus is loci.

# Simple loci

Example The locus of a point moves so that it is always a set distance (x) from a fixed point (O). What shape is it?

Imagine the minute hand on a clock. As the hand moves around the clock face, think of the path it follows.

Remember: the easiest way to draw this locus around a set point is to use a compass.

Example The locus of a point moves so that it is always a set distance (r) from the line between the points P and Q.

What shape is it?

The straight lines of the locus are parallel to the line from P to Q, because they are at a set distance (r) from the line. P and Q are fixed points at either end of the line, so we draw semicircles of radius r.

Sometimes the locus is not just a line, but an area. For example:

A cow grazing in the field ABCD moves so that it is always a distance of 5m from fence AB. Draw the locus of the cow.

# Constructions

## Perpendicular bisector of a line segment

The locus of a point which moves so that it is an equal distance from two points, A and B, is the perpendicular bisector of the line joining A and B.

Perpendicular means at right angles to.

Bisector means cuts in half.

To construct this locus, you do the following (try this yourself on a piece of paper):

Perpendicular bisector of a line segment

Draw the line segment XY.

Perpendicular bisector of a line segment

Put your compass on X and set it to be over half way along the line. Draw an arc.

Perpendicular bisector of a line segment

Without adjusting your compass put it on Y and draw another arc.

Perpendicular bisector of a line segment

Label these points A and B.

Perpendicular bisector of a line segment

Draw a straight line through A and B.

Perpendicular bisector of a line segment

The point M where the lines cross is the midpoint of XY. And AB is perpendicular to XY.

Practise this construction until you can do it without looking at the instructions. You don't need to label the points with letters each time - they are just there as a guide.

## Bisecting an angle

To bisect an angle, you do the following:

Bisecting an angle

V is the vertex of the angle we want to bisect.

Bisecting an angle

Place your compass on V and draw an arc that crosses both sides of the angle.

Bisecting an angle

Label the crossing points A and B.

Bisecting an angle

Place your compass on A and draw an arc between the two sides of the angle.

Bisecting an angle

Without adjusting your compass place it on B and draw another arc that cuts the one you just drew. Label the point where they cross C.

Bisecting an angle

Draw a straight line through V and C.

Bisecting an angle

The line VC bisects the angle. Angles AVC and BVC are equal.

Try this yourself on a piece of paper.

Remember: if you are asked to do a construction in an exam, do not rub out your construction lines.

## Equal distance from two lines

The loci of a point which moves so that it is the same distance from lines AOB and COD, are the lines which bisect angles AOC, AOD, DOB and BOC:

# Problems involving loci

Question

Two goats graze in a field of length 20m and width 12m. They are tethered to diagonally opposite corners by ropes of length 13m. On a scale drawing, show the area grazed by both goats.

Did you remember to use your compass? If so - well done!

If you did not get this right, here are some tips:

The question asks for a scale drawing. An appropriate scale would be 2m = 1cm. So on your drawing:

20m = 10cm

12m = 6cm

13m = 6.5cm

Do not forget to write the scale next to the diagram.

Next, measure a distance of 6.5cm using your compass. Place your compass point on one corner of the rectangle, and draw an arc. Do the same for the diagonally opposite corner.

The area between these 2 arcs is the area grazed by both goats.

# Drawing triangles

There are two methods of drawing triangles - construction or using a ruler and protractor.

## Construction

Construct a triangle with side lengths of 6cm, 5cm and 4cm.

Solution

1. Use a ruler to draw a 6cm line. Label one end A and the other B.
2. Open the compass to a radius of 5cm.
3. Place the compass needle at point A and draw an arc above the line.
4. Open the compass to a radius of 4cm.
5. Move the compass needle to point B and draw an arc above it.
6. Join each end of the line to the point where the arcs cross.

Remember, do not erase any construction arcs when using this method.

## Ruler and protractor example

Some triangles are better to draw using a ruler and protractor.

Draw a triangle with one 6cm side, and two angles of 55° 70°.

Solution

1. Use a ruler to draw a 6cm line. Label one end A and the other B.
2. Place the centre of the protractor at point A, taking care to line it up at zero degrees.
3. Measure an angle of 55° and mark it with a dot.
4. Join point A to the dot with a ruler.
5. Move the protractor to point B.
6. Measure an angle of 70° and mark the angle with a dot.
7. Join point B to the dot with a ruler.
8. If the lines don't meet use a ruler to make them longer until they do.

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