Pythagorean Theorem

Published on Sunday, September 02, 2007 in , , ,

right triangleDid I scare you with the title phrase? It's understandable. It brings up images of math class pressure during tests.

Could a math professor get something so simple as the Pythagorean Theorem wrong? In the math professor's class, where they are in control, probably not. However, if we replace the pressure of a test with the pressure of speaking on the radio, then it has actually happened. Peter Alfeld, of the Department of Mathematics at the University of Utah, admits to making a simple Pythagorean mistake on the radio.

The story may sound embarrassing, but it is nice to see people own up to their mistakes. By admitting to this mistake, however, Professor Alfeld does risk joining other notable people who had problems with the Pythagorean Theorem.

Probably the most famous instance of this is when the scarecrow from The Wizard of Oz got the Pythagorean Theorem wrong (or, more accurately, the script writers). This mistake was later parodied by no less than Homer Simpson, but at least someone was there to correct him:

Just so everyone is on the same page, the Pythagorean Theorem applies only to right triangles. It states that, given two legs of a right triangle, whose lengths are a and b, with a hypotenuse of length c, then a2 + b2 = c2.

The mistakes the fictional characters above was applying the formula to isosceles triangles (although the formula does hold true for isosceles right triangles). Professor Alfeld's mistake was different. The ladder against a wall did form a right triangle, but he neglected to realize that the ladder itself should have been the hypotenuse.

When students are tested on the Pythagorean Theorem, the questions usually provide the length of two of the sides, one of which may or may not be the hypotenuse, and ask you to find the length of the remaining side. However, what if you were only given the length of one of the legs (not the hypotenuse), and asked to find the lengths of the other two sides?

At first, this would sound impossible. If you're given a length of one leg as 8 inches, wouldn't there be too many possibilities for the other sides? Common examples would include 62 (36) + 82 (64) = 102 (100) and 82 (64) + 152 (225) = 172 (289).

There is a surprisingly simple way, given just the length of one leg (again, not the hypotenuse) to generate all the Pythagorean triples that involve only whole numbers or those ending in .5. I'll leave you with this slideshow that explains it quite well:

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