Pythagorean theorem

Elements and terminology

A right triangle has one 90° angle. The two sides adjacent to that angle are the legs and the side opposite is the hypotenuse, which is always the longest side. In practice the equality a^2 + b^2 = c^2 provides a test for perpendicularity: if the squared lengths satisfy this relation the triangle is right-angled (the converse of the theorem). The theorem is therefore a tool both for computing unknown lengths and for verifying orthogonality.

History and proofs

Although named for the Greek philosopher Pythagoras, versions of the rule appear in Babylonian and Indian mathematics and in ancient Chinese texts. The attribution to Pythagoras reflects early classical Greek development of proofs. Numerous demonstrations exist — geometric rearrangements, similarity arguments, algebraic derivations and area comparisons — and textbooks often present Euclid's geometric proof and several elementary dissections. One notable modern proof is attributed to James A. Garfield; overall the theorem has inspired many distinct proofs over the centuries.

Pythagorean triples and generation

Integer solutions (a,b,c) of a^2 + b^2 = c^2 are called Pythagorean triples. Many such triples exist, including the familiar (3,4,5) and (5,12,13). Primitive triples, whose entries share no common factor, can be generated by a simple parametric formula: for integers m>n>0, set a = m^2 - n^2, b = 2mn, c = m^2 + n^2; suitable choices of m and n produce all primitive triples. These integer solutions are a classical topic at the intersection of geometry and number theory.

Applications and generalizations

The theorem is widely used: in surveying and construction for computing heights and distances, in navigation and physics for resolving components, and in computer graphics and algorithms for computing Euclidean distances and testing orthogonality of vectors. In analytic form it connects with the dot product and the length of a vector in n-dimensional Euclidean space, where the squared length is the sum of the squares of the components. In non-Euclidean geometries the simple quadratic relation must be replaced by formulas that reflect curvature, and the law of cosines generalizes the relation to arbitrary triangles.

Examples and further reading

• Classic numerical example: 3^2 + 4^2 = 5^2, so a triangle with legs 3 and 4 has hypotenuse 5.
• Pythagorean triples can be used to construct integer-sided right triangles for practical and theoretical problems.
• Many elementary proofs rely on cutting and rearranging right triangles to form squares, illustrating the area interpretation of the identity.
• For more on the geometric context see articles on the right triangle and on alternative formulations in elementary treatment of triangle relations; for algebraic perspectives consult treatments of vector length and inner products in linear algebra.

The Pythagorean theorem remains central in mathematical education and a gateway to further topics in geometry, algebra and number theory. For introductory expositions and proofs see general references in elementary geometry and historical surveys that discuss the theorem's attribution and development across cultures (right-angle history, angle measurement discussions).