Section 84
The Pythagorean Theorem
The Pythagorean Theorem is named after Pythagoras 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.)
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 true only for right triangles – triangles which have a 90^{o} angle.
The converse of the theorem is also true: if a triangle has sides of lengths a, b, and c, and c^{2} = a^{2} + b^{2}, then it must be a right triangle.
PYTHAGOREAN TRIPLES
Three whole numbers a, b, c which satisfy the equation of the Pythagorean Theorem are called Pythagorean triples. A few of the smallest ones are shown in the table below. Each Pythagorean Triple corresponds with a right triangle whose side lengths are in wholenumber ratios.
Pythagorean Triples 
3, 4, 5 
3^{2} + 4^{2} = 5^{2}
9 + 16 = 25

6, 8, 10 
6^{2} + 8^{2} = 10^{2}
36 + 64 = 100

5, 12, 13 
5^{2} + 12^{2} = 13^{2}
25 + 144 = 169

8, 15, 17 
8^{2} + 15^{2} = 17^{2}
64 + 225 = 289
