Pythagorean Proof Based on the Principles of Scaling
Date: 04/04/2002 at 00:39:38 From: Charlene Subject: Innovative ideas for Pythagorean Theorem Hi Dr. Math, I'm a ninth-grader looking for help for a science fair project. I've decided to do a project with some connections to the Pythagorean theorem, but the project requires innovative ideas. The following are my current ideas for the project: a) Find several proofs of the theorem. b) Develop methods for finding pythagorean triples. (I already know of four, two of which I discovered myself) c) Investigate Fermat's Last Theorem. (It does have some connections with the Pythagorean Theorem, doesn't it?) Please, could you give me more ideas? Thank you very much for your help.
Date: 04/04/2002 at 00:55:31 From: Doctor Mitteldorf Subject: Re: Innovative ideas for Pythagorean Theorem I don't think you need more ideas, Charlene. You already have a bunch of good ones. You can start reading about the theorem in the Dr. Math FAQ: Pythagorean Theorem http://mathforum.org/dr.math/faq/faq.pythagorean.html Here are a few sites on the Web that prove the Pythagorean Theorem, many with pictures, some with more detail than others. Most involve areas in some form. Among these you should probably find at least one that is explained for you. Euclid's proof (with notes on two alternative proofs) - David Joyce http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html An Interactive Proof of Pythagoras' theorem (Euclid's proof in Java) - Jim Morey, UBC Mathematics Department http://sunsite.ubc.ca/LivingMathematics/V001N01/UBCExamples/Pythagoras/pythagoras.html A proof of the Pythagorean Theorem by Liu Hui (third century AD) - D. Wagner, University of Copenhagen http://www.staff.hum.ku.dk/dbwagner/Pythagoras/Pythagoras.html Pythagorean Theorem (Java applets) - Manipula Math http://www.ies.co.jp/math/java/geo/pythagoras.html 38 proofs and some more references - Cut-the-Knot, Bogomolny http://www.cut-the-knot.org/pythagoras/index.html The Pythagorean Theorem - Jim Loy http://www.jimloy.com/geometry/pythag.htm There are also several explanations in the Dr. Math archives, including these; the last proof Dr. Rob gives fits your description well, as does Dr. Mitteldorf's, which includes both picture and explanation: Proving the Pythagorean Theorem: A Traditional and a Modern Approach http://mathforum.org/library/drmath/view/54986.html The Pythagorean Theorem: A Modern Proof http://mathforum.org/library/drmath/view/54711.html And here's one that was new to me a few weeks ago. I like this proof because it seems to give you "something for nothing." There isn't any calculation in it, and it's based completely on the principles of scaling. Draw your right triangle with the hypotenuse as base. Let's call it ABC, with AB being the base and C being the point on top. Draw the altitude from C, making a right angle with the base. The altitude divides the triangle into two parts, so the areas of those two parts must add up to the total area. Now, it turns out that all three triangles - the big one and the two parts - are similar. "Similar" means they are the same shape, but different sizes. If all the angles match, then the triangles are similar. Can you prove that the angles match up in these three triangles? For the next step in the proof, you need a "scaling" principle. The principle says that if you take any shape and "scale it up" so that every line in it is f times bigger, then the area will be larger by f^2. To state the scaling principle in another way: the area of a figure is proportional to the square of any of its linear dimensions. So if any line in the figure has a length = d, then the area will be r*d^2, and the r will stay the same if you change d, so long as you change the whole figure proportionately. Now, the hypotenuse of the big triangle is c, and the hypotenuses of the parts are a and b. So the areas are rc^2 for the whole triangle, and ra^2 and rb^2 for the two parts. The parts add up to the whole, so ra^2+rb^2=rc^2. Then you can just divide both sides of the equation by r. Or, using the first statement of the scaling principle, if we take the big triangle as standard, with f = 1, then the "f" for one triangle is a/c and for the other it is b/c. The first scaling principle then tells us that (a/c)^2 + (b/c)^2 = 1. - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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