Difference between revisions of "Pythagorean Theorem"

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== Common Pythagorean Triples ==
 
== Common Pythagorean Triples ==
A [[Pythagorean Triple]] is a [[set]] of 3 [[positive integer]]s such that <math>a^{2}+b^{2}=c^{2}</math>, i.e. the 3 numbers can be the lengths of the sides of a right triangle.  Among these, the [[Primitive Pythagoren Triple]]s, those in which the three numbers have no common [[divisor]], are most interesting.  A few of them are:
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A [[Pythagorean Triple]] is a [[set]] of 3 [[positive integer]]s such that <math>a^{2}+b^{2}=c^{2}</math>, i.e. the 3 numbers can be the lengths of the sides of a right triangle.  Among these, the [[Primitive Pythagorean Triple]]s, those in which the three numbers have no common [[divisor]], are most interesting.  A few of them are:
  
 
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3-4-5

Revision as of 20:15, 4 November 2006

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The Pythagorean Theorem states that for all right triangles, ${a}^{2}+{b}^{2}={c}^{2}$, where c is the hypotenuse, and a and b are the legs of the right triangle. This theorem is a classic to prove, and hundreds of proofs have been published. The Pythagorean Theorem is one of the most frequently used theorem in geometry, and is one of the many tools in a good geometer's arsenal.

This is generalized by the Pythagorean Inequality (See geometric inequalities) and the Law of Cosines.)


Introductory

Example Problems

Common Pythagorean Triples

A Pythagorean Triple is a set of 3 positive integers such that $a^{2}+b^{2}=c^{2}$, i.e. the 3 numbers can be the lengths of the sides of a right triangle. Among these, the Primitive Pythagorean Triples, those in which the three numbers have no common divisor, are most interesting. A few of them are:

3-4-5

5-12-13

7-24-25

8-15-17

9-40-41

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