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### The Pythagorean Theorem: A Modern Proof

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Date: 04/14/97 at 05:24:34
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?
```

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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/
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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