## Pythagoras’ Theorem works with Negative Length

When taking square roots of both sides of an equation, one should be careful not to turf out the negative result without first considering whether it has a true meaning. When using Pythagoras’ Theorem, the last step is to take square roots. So, can we have a hypotenuse with length -5?

I taught Pythagoras’ Theorem this week to my year 8 girls. Many are exceptional mathematicians for their age and it’s very important they get into good habits early, so, when we took square roots at the end, I reminded them that there are two results, one positive and one negative, but we reject the negative because length can’t be negative.

**Or can it?**

**In the interactive diagram below, click and drag vertex B up and down. **

Watch the numbers at the bottom. Due to the Theorem, the area of the green square is equal to the sum of the areas of the red squares, wherever vertex B is (as long as it’s a right-angled triangle). It transpires that this is also true if you drag vertex B below vertex C, thus turning the triangle inside-out. The vertical length AC becomes negative, but when we square it we still get a positive area, and the Theorem holds.

Pythagoras’ Theorem is incredibly resilient! This might explain why it’s lasted 2500 years.

**Drag vertex B up and down**

So, regarding my question: “Can we have a hypotenuse with length -5?” The truth is, I’m not sure. But we can definitely have a *shorter side* with negative length because that’s what the diagram shows.

If you figure out how to make the hypotenuse negative, do let me know.

The hypotenuse is declaratively positive, but it can arguably be deductively negative as well. The web address http://www.issuu.com/tesseractdweller stages a strong case for the deductive isolation of a negative scalar distance value between two points. Since two points so specified can also delineate any of a set of segments that comprise a radius, there wouldn’t be one segment in such a set that isn’t subject to sufficient negative delineation. The hypotenuse in trigonometric functions is essentially a radius when also a unit segment having one point in common with the origin of two rectangular coordinate axes: The hypotenuse would, therefore, be just as eligible to be negative as the radius is.