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Difference between revisions of "Pythagorean Theorem"

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The '''Pythagorean Theorem''' states that for all [[right triangle|right triangles]], <math>{a}^{2}+{b}^{2}={c}^{2}</math>, 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. Many proofs of it don't have any words and are simply pictures that challenge the reader to figure out how the picture works. 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. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem.  
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The '''Pythagorean Theorem''' states that for a [[right triangle]] with legs of length <math>a</math> and <math>b</math> and [[hypotenuse]] of length <math>c</math> we have the relationship <math>{a}^{2}+{b}^{2}={c}^{2}</math>. This theorem has been know since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually. 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. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem.  
  
 
This is generalized by the [[Pythagorean Inequality]] (See [[geometric inequalities]]) and the [[Law of Cosines]].)
 
This is generalized by the [[Pythagorean Inequality]] (See [[geometric inequalities]]) and the [[Law of Cosines]].)

Revision as of 11:22, 10 November 2006

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The Pythagorean Theorem states that for a right triangle with legs of length $a$ and $b$ and hypotenuse of length $c$ we have the relationship ${a}^{2}+{b}^{2}={c}^{2}$. This theorem has been know since antiquity and is a classic to prove; hundreds of proofs have been published and many can be demonstrated entirely visually. 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. A very large number of geometry problems can be solved by building right triangles and applying the Pythagorean Theorem.

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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