Pythagoras theorem

 Pythagorean theorem, the well-known geometric theorem that the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse (the side opposite the right angle)—or, in familiar algebraic notation, a² + b² = c². Although the theorem has long been associated with Greek mathematician-philosopher Pythagoras (c. 570–500/490 BCE), it is actually far older. Four Babylonian tablets from circa 1900–1600 BCE indicate some knowledge of the theorem, with a very accurate calculation of the square root of 2 (the length of the hypotenuse of a right triangle with the length of both legs equal to 1) and lists of special integers known as Pythagorean triples that satisfy it (e.g., 3, 4, and 5; 3² + 4² = 5², 9 + 16 = 25). The theorem is mentioned in the Baudhayana Sulba-sutra of India, which was written between 800 and 400 BCE. Nevertheless, the theorem came to be credited to Pythagoras. It is also proposition number 47 from Book he length of any side of any triangle, 7 the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation

If c denotes the length of the hypotenuse and a and b denote the two lengths of the legs of a right triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation:

${\displaystyle a^{2}+b^{2}=c^{2}.}$

If only the lengths of the legs of the right triangle are known but not the hypotenuse, then the length of the hypotenuse can be calculated with the equation

${\displaystyle c={\sqrt {a^{2}+b^{2}}}.}$

If the length of the hypotenuse and of one leg is known, then the length of the other leg can be calculated as

${\displaystyle a={\sqrt {c^{2}-b^{2}}}}$

or

${\displaystyle b={\sqrt {c^{2}-a^{2}}}.}$

A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

Pythagorean theorem