Section 12-4
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:
c2 = a2 + b2
(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 90o angle.
The converse of the theorem is also true: if a triangle has sides of lengths a, b, and c, and c2 = a2 + b2, 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 whole-number ratios.
Pythagorean Triples |
3, 4, 5 |
|
6, 8, 10 |
62 + 82 = 102
36 + 64 = 100
|
5, 12, 13 |
52 + 122 = 132
25 + 144 = 169
|
8, 15, 17 |
82 + 152 = 172
64 + 225 = 289
|
