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Wednesday Math, Vol. 102: The Pythagorean Theorem revisited

Since this is my 102nd weekly math post, which doesn't count posts that I put in that include mathy-ness that aren't published on Wednesdays, it should come as no surprise that I've written about the Pythagorean Theorem before, way back in Volume 3 of Wednesday Math. Last week, I talked about base 60 being ancient math, borrowed from the Sumerians and still in use when we deal with seconds and hours and degrees in geometry. The Pythagorean Theorem is likewise ancient, and it was understood by civilizations around the world and proven many times in many different ways. In my first post on the Theorem, I presented two proofs that deal with the areas of squares and the areas of right triangles to prove a² + b² = c². The only math we need to make these types of proofs work is the ability to expand (a + b)² and the knowledge that in a right triangle, the simplest base and height are the two short sides a and b, often referred to as the legs of the triangle so the area can be written as ½*ab. People are more likely to remember the word hypotenuse, the special name for the long side of a right triangle.

Here's a completely different proof using different math to prove the same thing. The picture is a right triangle laid flat on the hypotenuse with a square below it, which means the area of the square is c². A vertical line is drawn from the top the triangle to the base of the square, cutting the square into two rectangles. The drawing is much simpler than other Pythagorean proofs, but the math is a little trickier. The strategy is to find the areas of the two rectangles that make up the square and prove that have areas equal to a² and b², so the area of the square they combine to form can be written either as c² or a² + b². Instead of the simple math of the area of squares and right triangles being used, we have to use the rules of similar triangles to complete this proof. Here is the previous diagram labeled. Our goal is to re-write the areas of the rectangles PSUV and RTUS, which we know are cx and cy, respectively. The three triangles, QPR, SQR and SPQ, are all right triangles, and since the little triangles share an angle with the big triangle that is not the right angle, each of them must be similar to the big triangle, which means they are similar to each other. There are many properties of similar triangles, but the one we need here is that if we line up the sides in the correct order, the ratios between corresponding sides of two similar triangles are equal. This is a fancy way of saying that geometrically similar objects are effectively "the exact same shape" drawn to different scales. (Technically, congruent shapes, same shape and same scale, are still similar, so the scales don't have to be different, but in this drawing, all three triangles are different sizes.) We are going to use these facts to write x and y in terms of a, b, and c.

The lengths of the sides of QPR and SQR in the same order.

Sides of triangle QPR in ascending order of length: a b c
Sides of triangle SQR in ascending order of length: y h a

We set up the ratio equality marked in bold, a/y = c/a. Cross multiplying we get a² = cy, which is to say the slender rectangle on the right has the same area as the square of the short leg of the right triangle. Now we take the same steps with the triangles QPR and SPQ, again with the relevant sides marked in bold.

Sides of triangle SPQ in ascending order of length: h x b

When we take the ratio b/x = c/b and cross-multiply, we get cx = b². This proves the rectangles are equal in area to the squares of the lengths of the legs, squares we did not even bother to draw in our diagram because they were not part of the proof. I blush to say that I only discovered the existence of this proof a few weeks ago. If you are one of those weirdos who really like math, this should be right up your alley. One of the great beauties of math is that completely different methods can be used to get to the same place, and learning those different paths can show you that things you might have considered unrelated to one another share some deep, underlying truth.

About two years ago, I wrote in passing that a really excellent proof is as pretty as a hummingbird. That's certainly how feel about this Pythagorean proof and others, and the added marvel of how different they are from each other.
Posted by Matthew Hubbard at 3:09 AM Labels: Math 
That's a very nice proof indeed.

An unfortunate font choice (or maybe it's my monitor) makes the 'y' look a lot like a 'v', or possibly a nu.

December 23, 2009 at 10:14 AM 
If you are talking about the 'y' in the second illustration, you are right, it's a little funky.

You are also right that it's a nice proof.

December 23, 2009 at 1:27 PM 

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