Pythagorean Theorem

The Theorem

In a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

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

Where:

• $a$ and $b$ are the legs
• $c$ is the hypotenuse (longest side, opposite the right angle)

Using the Theorem

To find the hypotenuse: $c = \sqrt{a^2 + b^2}$

To find a leg: $a = \sqrt{c^2 - b^2}$

Pythagorean Triples

Sets of three positive integers that satisfy $a^2 + b^2 = c^2$:

Common triples:

• $3, 4, 5$
• $5, 12, 13$
• $8, 15, 17$
• $7, 24, 25$

Any multiple works too: $6, 8, 10$ (multiply $3, 4, 5$ by 2)

Converse

If $a^2 + b^2 = c^2$, then the triangle is a right triangle.

Distance Formula

The Pythagorean Theorem leads to the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$