230hw1sol - Math 230 E Fall 2011 Homework 1 Drew Armstrong...

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Unformatted text preview: Math 230 E Fall 2011 Homework 1 Drew Armstrong Problem 1. In the Lilavati , the Indian mathematician Bhaskara (1114–1185) gave a one-word proof of the Pythagorean theorem. He said: “Behold!” Add words to the proof. Your goal is to persuade a high school student who claims he/she doesn’t “get it”. Try to avoid algebra. (Sorry, the two pictures are not quite to scale.) I will present two solutions. In both we label the short, medium, and long sides of the triangle by a , b , and c , respectively. Solution 1. This is the solution that Bhaskara had in mind. There are three important observations: (1) The two figures have equal area because they are made out the same five pieces; i.e. four copies of the triangle and one square of side length b- a . (2) The figure on the left is a square of area c 2 . (3) The figure on the right is divided by the dotted line into two squares, the smaller of area a 2 and the larger of area b 2 . (To convince your high school pupil of this you should probably add labels to the right figure.) Combining the observations we conclude that c 2 = a 2 + b 2 . Solution 2. This solution is modern, and it ignores the right figure. This is definitely not what Bhaskara had in mind. Note that there are two ways to compute the area of the left figure. On one hand, it is a square of area c 2 . On the other hand, it is formed from a square of area ( b- a ) 2 and four triangles of area ab/ 2. We conclude that c 2 = ( b- a ) 2 + 4( ab/ 2) = a 2 + b 2-...
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This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.

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230hw1sol - Math 230 E Fall 2011 Homework 1 Drew Armstrong...

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