# Difference of two line segments

This construction shows how to create a line segment whose length is the difference between two given segments. This is very similar to Sum of line segments, except that the second segment is drawn on the other side of the first, effectively subtracting the lengths.

The two types of construction (add , subtract) can be combined in any way you like. For example you could find the resulting length of a+b–d–f.

## What if the second segment is longer than the first?

If b is longer than a, then you will find the end of the construction to the left of the start: The distance PQ is still the difference between the two segment lengths. In arithmetic, this would result in a negative length, which does not exist in Euclidean geometry. Segments cannot have a negative length.

So the precise definition of what this construction does is find the absolute value of a–b The two vertical bars mean "absolute value" which is always positive, regardless of the way a–b comes out. This is why the construction is titled "Difference of two segments" – it is not quite the same as arithmetic subtraction.

## Proof

The proof of this construction is trivial. This is the same drawing as the last step in the above animation. Argument Reason
1 The segment PQ is congruent to the given segment a Copied using the procedure in Copying a line segment. See that page for proof.
2 The segment RQ is congruent to the given segment b As in (1)
4 PR is the difference of given segments a,b,c From (1), (2). All segments are colinear and adjacent.

- Q.E.D

## Printable step-by-step instructions

The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.

## Try it yourself

Click here for a printable worksheet containing two line segment copying problems. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.