Relationship between the lengths

*a*,*b*and*c*of the sides of a right triangle such that*a*, where^{2}= b^{2}+ c^{2}*a*represents the length of the hypotenuse and*b*the lengths of the two sides of the right angle.This is a concrete representation of the use of the Pythagorean theorem, where : \({2}^{2} + {3}^{2} = (\sqrt{13})^{2}\)

A **Pythagorean triangle** is a right triangle whose side lengths are expressed as whole numbers.

The triple of numbers obtained is a Pythagorean triple.

These measures can be determined by choosing two whole numbers *m* and *n*, with *m* > *n*, and applying the following relationship :

*a*=*k*(*m*\(^{2}\) –*n*\(^{2}\))*b*=*k*(2*mn*)*c*=*k*(*m*\(^{2}\) +*n*\(^{2}\))- Pythagorean triples can be determined by replacing
*k*with the sequence of whole numbers.

If*m*= 5,*n*= 2 and*k*= 3, we obtain :*a*= 42,*b*= 40 and*c*= 58.

This can be verified as follows : \(42^{2} + 40^{2} = 58^{2}\) and 1764 + 1600 = 3364.

### Historical note

Although this relationship was known before the time of Pythagoras and his school of thought, it bears his name in various works that came after him to acknowledge that it was studied and applied in Pythagoras’s disciplines.

In the 18th century, students called this relationship the *Bridge of Asses*.