Ruler and Compass Constructions

In this assignment we will learn how to do several constructions using only a ruler for drawing straight lines and a compass for drawing circles. We will not need the ruler for measuring distances. Determining how to perform constructions was a major component of the study of geometry for the ancient Greeks, and it continues to be a component of geometry today. In addition, these constructions will allow us to make precise constructions. For instance, the procedure below for constructing a regular hexagon generally gives a better hexagon than if one uses a protractor to draw angles.

The ancient Greeks had the following rules for making constructions. Given two points, one can construct the line passing through them and given two points, one can construct the circle centered at one point and passing through the other. Furthermore, one can construct a point if it is on the intersection of two lines, two circles, or a line and a circle. No other constructions are allowed. Recall that a line goes on forever. Therefore, we can only draw part of it, which is called a line segment. If we want to indicate a line, we put arrows at the ends to indicate it keeps going on.

Constructing a Perpendicular Bisector

Given a line segment AB, here is a method to construct the perpendicular bisector.

  1. Draw the circle centered at A and passing through B.
  2. Draw the circle centered at B and passing through A.
  3. The two circles intersect at two points. Draw the line through these two points. This line is the perpendicular bisector to AB.

Constructing an Angle Bisector

Given an angle, call the vertex of the angle point A, here is a method to construct the angle bisector.

  1. Draw a circle centered at A. Its size does not matter, as long as the circle crosses both sides of the angle. Label the points B and C where the circle crosses the angle.
  2. Draw the circle centered at B and passing through A. Also draw the circle centered at C and passing through A. You only need to draw enough of these circles to see where they intersect. The less you draw the less cluttered your picture will be.
  3. The two circles intersect at a point (other than A). Draw the line through this intersection point and A. This line bisects the angle.

Constructing an Equilateral Triangle

Draw a line segment. Take your compass and spread it apart to be the length of the segment. Draw a circle centered at one endpoint (and passing through the other). Then draw another circle centered at the other endpoint with the same radius. Take one of the two points where the two circles intersect. This will be the third vertex of an equilateral triangle; the two endpoints are the other two vertices.

Constructing a Regular Hexagon

Draw a circle and mark one point on it. Keeping the compass spread out the same amount, place the compass at the point you marked on the circle. Draw enough of the circle to see where it crosses the original circle, and mark the point of intersection. Next, place the compass on this new point and repeat this process to obtain a third point. Keep doing this until you have six points. Connect them to obtain the hexagon.

In the problems below, describe your construction in such a way that somebody will be able to take your procedure and recreate your construction. If you use a previous construction, say so and be specific about how you use it, but do not write down its steps.

Problem 1. Starting with a line and a point on the line, describe how to construct the line that passes through the point and is perpendicular to the given line.

Problem 2. Starting with a line and a point not on the line, describe how to construct the line that passes through the point and is parallel to the given line.

Problem 3. Starting with a line segment, describe how to construct a square one of whose sides is the line segment.