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The Pythagorean Theorem

ThePythagorean Theoremis named afterPythagoras of Samos, a mathematician who was also a religious leader, and believed that all things in the universe were composed of numbers.

He is supposed to have been the first to have proved this theorem about right triangles:

Pythagorean Theorem.In a right triangle with legs of lengths$a$and$b$and hypotenuse of length$c$, the following equation is true:

$c^2=a^2+b^2$

(There are many different ways to prove this.)

Thehypotenuseof a right triangle is the side opposite the right angle.

Below is a graphical representation. The theorem states that the sum of the areas of the blue and red squares is equal to the area of the green square.

Important: Remember that the Pythagorean Theorem is trueonlyfor right triangles – triangles which have a$90°$angle.

Theconverseof the theorem is also true: if a triangle has sides of lengths$a,b,\text{and}c$, and$c^2=a^2+b^2$, then it must be arighttriangle.

See alsoPythagorean Triples.