The Pythagorean Theorem: A Modern Proof

Date: 04/14/97 at 05:24:34
From: Rosalyn Adams
Subject: Proving the Pythagorean Theorem

I am currently doing research on the Pythagorean Theorem for my 
Algebra II/Trig. class. I've looked at a lot of math Web sites, but 
none seems to answer my question. I know that the Pythagorean Theorem 
works and I can show how it works, but what I  really need to know is 
why it works.

Also would you have any ideas for my project, which consists of 
explaining the theorem and its important parts using mathematical 
language and symbolism?

Date: 04/14/97 at 10:39:55
From: Doctor Mitteldorf
Subject: Re: Proving the Pythagorean Theorem

Dear Rosalyn,

Pythagoras's proof of his theorem is rather hard to follow, but there 
is a modern proof that is much easier.  It does involve several lines 
of algebraic manipulation, which you must be prepared to do:

Draw a horizontal line which you'll divide into two parts, labeled 
a and b.  Now draw a vertical line at each side, and another 
horizontal line on the top, so you have a square of side a + b.

Divide your square into four sections.  There's an a^2 in  one corner, 
a b^2 in the diagonally opposite corner, and two side-rectangles of 
area a*b.

           

Now start over again and re-draw the same square, but split the sides
differently.  Each line going around the square clockwise should be
divided long-short, long-short, long-short, long-short.  Now when you 
connect the 4 dividing points on each line segment, you can make a
diagonal square in the middle.  In other words, your figure consists 
of a diagonal square whose side is the hypotenuse c, surrounded by 
four a-b-c right triangles.   

           

Now if you express the area of the big square in two different ways 
and clear out the algebra, you will be left with the Pythagorean 
theorem.

Please spend a little time with it - try to make sense out of it on 
your own, and work with the algebra aiming for a^2 + b^2 = c^2.  

If you still have trouble, please write back.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/