Garfield's Pythagorean Theorem ProofDate: 02/27/97 at 23:10:00 From: Mr. Michael D. Ferris, Sr. Subject: Pythagorean Theorem Dear Mr. Math: I am a ninth grader at Tupelo High School (Tupelo, Mississippi), and my Accelerated Geometry Teacher has assigned the following question for extra credit: Which President of the United States wrote a proof of the Pythagorean Theorem? On what did he base his proof? Give examples and provide copy of proof. We have searched the Internet and the library at school and have been unable to find information concerning these questions. Thank you so much! Date: 02/28/97 at 12:14:57 From: Doctor Wilkinson Subject: Re: Pythagorean Theorem The president was James Garfield, and this is the proof: Take two copies of your right triangle with sides a, b, c. Place one of them with side b horizontal and side a vertical with the right angle at the lower left. Place the second triangle with its side b as a continuation of the side a of the first triangle and with the right angle at the upper left. Join the other vertices of the two triangles. This give you a trapezoid with upper base a, lower base b, and height a + b. Now we're going to compute the area of the trapezoid in two different ways. First we have the formula A = h * (a + b) / 2 In this case h = a + b, so we get A = (a + b)^2 / 2 On the other hand, the trapezoid is made up of three triangles. Two of these are the two copies of the original right triange. The third is the isosceles right triangle with two sides equal to c. The sum of the areas of these is (1/2) ab + (1/2) ab + (1/2) c^2 If you equate the two formulas you get the Pythagorean Theorem. Pretty clever, isn't it? -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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