In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states: In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle). The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides. These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations. Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics. The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including extension to many-dimensional Euclidean spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, although it is often argued that knowledge of the theorem predates him. (There is much evidence that Babylonian mathematicians understood the formula, although there is little surviving evidence that they fitted it into a mathematical framework.) “[To the Egyptians and Babylonians] mathematics provided practical tools in the form of "recipes" designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right.” In addition to a separate section devoted to the history of Pythagoras' theorem, historical asides and sources are found in many of the other subsections. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power. The article ends with a section on pop references to the theorem.
The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). -------------------------------------------------
As pointed out in the introduction, if c denotes the length of the hypotenuse and a and b denote the lengths of the other two sides, Pythagoras' theorem can be expressed as the Pythagorean equation:
or, solved for c:
If c is known, and the length of one of the legs must be found, the following equations can be used:
The Pythagorean equation provides a simple relation among the three sides of a right triangle so that if the lengths of any two sides are known, the length of the third side can be found. A generalization of this theorem is the law of cosines, which allows the computation of the length of the third side of any triangle, given the lengths of two sides and the size of the angle between them. If the angle between the sides is a right angle, the law of cosines reduces to the Pythagorean equation. -------------------------------------------------
This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs. Proof using similar triangles
Proof using similar triangles
This proof is based on...