Pythagorean theorem

Animated gif version of SVG of rearrangement proof of Pythagorean theorem

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Pythagoras similar triangles simplified

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Illustration to Euclid's proof of the Pythagorean theorem

Illustration to Euclid's proof of the Pythagorean theorem2

Pythagorean Theorem is a fundamental principle in Euclidean geometry that establishes a relationship between the lengths of the sides of a right triangle. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be expressed by the equation:

\[a^2 + b^2 = c^2\]

where c represents the length of the hypotenuse, and a and b represent the lengths of the triangle's other two sides.

History

The theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery and proof, although evidence suggests that the knowledge of the theorem predates him. The Babylonians, Indians, and Chinese were aware of this theorem long before Pythagoras; however, Pythagoras is often credited due to the geometric proof attributed to him.

Proofs

Over the centuries, numerous proofs of the Pythagorean Theorem have been devised. These include geometric, algebraic, and dynamic proofs. One of the most famous geometric proofs involves rearranging two squares that have been divided into triangles and squares, demonstrating the relationship in a visual manner.

Applications

The Pythagorean Theorem has widespread applications in various fields such as mathematics, physics, engineering, and architecture. It is used in calculating distances, in trigonometry, in the construction of buildings, and in navigation. The theorem is also fundamental in the derivation of the distance formula in Cartesian coordinates.

Generalizations

The Pythagorean Theorem has been generalized in several ways. One notable generalization is the Law of Cosines, which extends the theorem to non-right triangles. Additionally, the theorem has analogs in non-Euclidean geometries, such as spherical and hyperbolic geometry, where it takes a different form due to the curvature of the space.

See Also

• Euclidean geometry
• Triangle
• Hypotenuse
• Mathematics in ancient Greece