# Loci and Constructions

A locus is just a line or region that shows all the points that fit a given rule. There’s a few types.

Loci That Are Fixed Distances From A Given Point

If something is a fixed distance from a given point then it makes a circle and you can draw it with a compass. Loci That Are A Fixed Distance From A Given Line

These have a straight and a curved bit. Draw the straight bit with a ruler and curved bit with a compass.

E.g. below shows a locus that is a fixed distance from line AB. Loci That Are Equidistant From Two Given Lines

Here you need to bisect an angle.

Let’s take an example. Say you had to bisect the angle made by the line PQR, here is what you would do: Place your compass on Q and make a mark on the lines. Call these A and B.

Then keeping your compass set at the same length draw two further marks. Mark point C where they intersect.

Draw from a straight line from Q going through C.

You have now cut the angle in half.

Loci Equidistant From Two Points

Here you need to draw the perpendicular bisector. This sounds complicated but all it means is it cuts something in half and is at right angles to it.

Let’s take an example. Say you had to cut a line AB in half, here’s what you would do: Place your compass on A and set it to over half the distance of the line. Draw a mark above and below the line. Do the same from point B, keeping the compass set at the same length.

Draw a line between the points where the marks meet.

NOTE: This is the correct method for drawing an angle of 90° without a protractor. If you needed a 45° angle you could just bisect this angle.

Constructing Angles of 60°

To do this draw a line AB. Then putting your compass on point A make a big arc. Then, keeping your compass at the same length, place it on the point where the arc crosses line AB. Now draw another arc. Draw a line between A and where the 2 arcs cross…and that’s it! You might need to use this method if asked to draw an equilateral triangle without a compass!

Constructing A Perpendicular From A Line

This is similar to the perpendicular bisector method above…but this time you’re given a point and a line and the method is slightly different.

In the example below the point is R. You would put your compass on the point and make two marks on the line. Then place your compass, set at the same length, on each of these points and make two more marks. Then join R to where these marks meet. 