Pythagorean and Other Triangle Theorems

Date: 6 Jan 1995 10:46:47 -0500
From: Neel Murarka
Subject: (none)

        Could you please show me the proof for the Pythagorean Theorem
and list the other triangle theorems that are based on it?  Thank you.

Bmurarka@walrus.mvhs.edu
:-)(-:

Date: 10 Jan 1995 04:33:25 -0500
From: Dr. Sydney
Subject: Re: your mail

Dear Neel, 

        Thanks for writing to Dr. Math!  There are actually many proofs of
the Pythagorean Theorem, so I'll just show you one of the most famous 
proofs which was done by a mathematician named Euclid who lived 
around 300 B.C.E.

        I'm going to attempt a picture, here:

            H            G
                           
                   C   
                  / \
        K        /   \        F
                /     \ <--- this is supposed to be a right triangle
               A----x'-B     with the right angle at the top.
               |       |
               |       |
               |       |
               D----x--E

Okay, that didn't work so well.  Let me tell you, it is quite challenging
drawing on a computer.  I'll explain what I was trying to draw.  First,
you draw your right triangle (ACB in my picture, with the right angle at C),
and then you draw squares along each side of the triangle.  So, side AC has
a square sticking out of it with sides of the same length as side AC.  The
other verticies of the square are K and H in my picture (sorry, I couldn't
draw in the lines!).  Likewise there is a square with sides the same length
as side CB -- the vertices of this square are G and F, and there is a square
with sides the same length as AB whose vertices are D and E.  Now drop a
perpendicular line from C to the line CE.  The point where this line
intersects AB is X' and the point where this line intersects CE is X. 

        Okay, so now, you should have a picture.  The basic idea of the
proof is to show that the sum of the areas of the two smaller squares (the
ones with side lengths equal to side lengths of the legs of the triangle) is
equal to the area of the biggest square.  If we show this is true, we've
proven the Pythagorean Theorem. Do you see why?

        So, let's get to it.  I don't want to go through it in great detail
because that would take a lot of time, but I'll give you the basic steps,
and you can try to figure out the reasons for the steps. If you have any
problems with the reasoning behind the steps, please feel free to write
back.  

        Okay, here we go!

        First show the area of the square with length AC is twice the area of
the triangle ABK.  Second, show the triangles ABK and ACD are 
congruent. Third, show the area of rectangle ADXX' is twice the area of 
triangle ACD, thus the area of the square with sides the length of AC is 
equal to the area of rectangle ADXX'.  

        Follow similar steps to show that area of the square with sides the
length of BC is equal to the area of rectangle BX'XE.  Then that means that
the sums of the areas of the two squares with sides of length AC and BC is
equal to the area of the square with side length AB.

        I'm not sure how much or how recently you've had geometry, but the
reasoning should follow from a high school geometry course.  I would 
suggest drawing a picture, labeling everything, and then trying the steps of
Euclid's proof.  Write back if you get stuck somewhere!

        As for your other question about other triangle theorems that are
based on the Pythagorean Theorem, I'm not sure quite what to say, because
the Pythagorean Theorem is used in so many ways!  One extremely helpful 
way it is used is in trig to get the identities: cos^2 x + sin^2 x = 1, etc.
But, there are many, many problems and theorems that use the result of the
Pythagorean Theorem.  It is indeed quite significant!  

        I hope this helps some.  Write back if you have any other questions. 

--Sydney, Dr. "heh, heh" math!