Geometrical Constructions

Introduction Making accurate drawings is not a skill confined to artists. Draftsmen, architects, engineers, and designers need to be able to construct geometrical figures.

In this unit we will look at how to carry out some common geometric constructions using only a pair of compasses and a ruler.

Equidistance

In the map shown in Fig.1 below, the town of Richley is 5 km from both Bridgeford and from Adleborough. We say that Richley is equidistant from Bridgeford and from Adleborough.

Figure 1. Three towns marked on a map.

When we are told only that a place is say 8 km away from us we do not know in what direction it lies, so we do not know its exact position. However, from this information we can find all the possible positions of the place. If we plot a

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 the same distance away in every possible direction, we find we have drawn a circle. Click somewhere in the animation space below to set a distance from the starting point and see what happens.

Figure 2. The possible positions of a point a given distance away.

If you were told that Ceyton was

equidistant

Items that are at an equal distance from an identified point, line or plane are said to be equidistant from it.

equidistant from Ayton and Beeville, and you were given the following map, can you see where the possible positions of Ceyton would be? The circles represent all the points a given distance from Ayton and all the points the same distance away from Beeville. Try dragging the green handle to alter the size of the red circles (they should always be the same size as one another).

Figure 3. Map of the Ayton area.