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12.5 Drawing a locus

12.5 Drawing a locus

A locus is a set of points that follow a given rule. $e plural of locus is loci.
You need a ruler and compasses to draw loci accurately.

Worked example 12.5

a A is a fixed point. Draw the locus of points that are all 2 cm from A.
b Draw the locus of points that are exactly 3 cm from:
i a given line ii a given line segment BC.

a Use compasses to draw a circle of radius 2 cm with centre A.
All the points on the circle are 2 cm from A, so this is the required
A locus of points.
2 cm All the points inside the circle are less than 2 cm from A. All the
points outside the circle are more than 2 cm from A.

b i Using a ruler, on either side of the given line, measure and mark two
3 cm 3 cm
points that are 3 cm from the line. Then join the points on each side
of the given line to draw two parallel lines. These lines, shown in red,
3 cm 3 cm
are the required locus of points exactly 3 cm from the line.
A straight line has infinite length, so the locus of points also has
infinite length.

ii The line segment BC has endpoints at B and C. Follow the steps in
3 cm 3 cm C part i to draw the parallel lines either side of the line segment. From
B 3 cm
3 cm points B and C draw semicircles of radius 3 cm to complete the locus
of points, shown in red, that are 3 cm from the given line segment.
) Exercise 12.5

1 The diagram shows a point P. Copy the diagram on plain paper. P
Draw the locus of points that are exactly 5 cm from point P.
2 Draw a straight, horizontal line.
Draw the locus of points that are exactly 2 cm from the line.
3 Draw a line segment AB, 6 cm long. A B
Draw the locus of points that are exactly 3 cm from AB. 6 cm

4 A donkey is tied by a rope to a post in a field. The rope is 8 m long.
Draw the locus of points that the donkey can reach when the rope is tight.
Use a scale of 1 cm to 2 m.

5 A coin has radius 1.5 cm. C
The coin is rolled around the inside of a rectangular box, so
that it is always touching a side of the box. 1.5 cm 10 cm
The box measures 10 cm by 12 cm.
Draw the locus of C, the centre of the coin.
12 cm

12 Tessellations, transformations and loci 121