Name
Many people think Pythagoras discovered the "Pythagorean Theorem", or at least provided the first proof of it. Neither of these is true, Pythagoras learned of the theorem, or its converse, from the Babylonians (Iraqi's) and had no reason to prove it, at least not in the modern sense of a proof. Pythagoras was interested in whole numbers (positive integers) and the theorem was named after him because he was the first to find many different triples of whole numbers (a, b, c) with a2 + b2 = c2.
a | b | c | b + c | a2 | b2 | c2 |
---|---|---|---|---|---|---|
3 | 4 | 5 | 9 | 9 | 16 | 25 |
5 | 12 | 13 | 25 | 25 | 144 | |
7 | 24 | 25 | 49 | 49 | ||
9 | 40 | 41 | 81 | |||
11 | 60 | 61 | ||||
13 | 84 | |||||
15 |
Problem 2 Find the pattern and complete the chart below. In the last column give a decimal (calculator) value for d / s.
s | d | s2+ s2 | d2 | d / s |
---|---|---|---|---|
1 | 1 | 2 | 1 | 1.0 |
2 | 3 | 8 | 9 | 1.5 |
5 | 7 | 50 | 49 | 1.4 |
12 | 17 | 288 | 1.41666667 | |
29 | 41 | 1681 | ||
70 | ||||
Given that the square root of 2 is approximately 1.4142135 what do you think is happening to d / s? What would this ratio be in a "real" right triangle?
Suppose there is an isosceles right triangle with integer sides. Among all such triangles there must be one with smallest sides (why?). Suppose this triangle had side S and hypotenuse D. Then let s = D - S and d = 2S - D, the isosceles triangle with equal sides s and third side d is also an isosceles right triangle (why? give an algebraic or geometric proof). But s,d are also integers and are smaller than S, D so our original supposition could not have been true.