The Pythagorean Theorem

One of the most important and useful theorems in all of mathematics!

The Theorem

For any right triangle:

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

Where:

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

Visual Understanding

The theorem says: The square built on the hypotenuse has the same area as the two squares built on the legs combined.

Finding the Hypotenuse

Given the legs, find the hypotenuse.

Example: Legs are 3 and 4. Find the hypotenuse.

$c^2 = 3^2 + 4^2$ $c^2 = 9 + 16$ $c^2 = 25$ $c = 5$

Finding a Leg

Given the hypotenuse and one leg, find the other leg.

Example: Hypotenuse is 13, one leg is 5. Find the other leg.

$a^2 + 5^2 = 13^2$ $a^2 + 25 = 169$ $a^2 = 144$ $a = 12$

Pythagorean Triples

These are sets of whole numbers that satisfy the theorem:

Triple	Check
3, 4, 5	$9 + 16 = 25$ ✓
5, 12, 13	$25 + 144 = 169$ ✓
8, 15, 17	$64 + 225 = 289$ ✓
7, 24, 25	$49 + 576 = 625$ ✓

Multiples also work: 6, 8, 10 (double of 3, 4, 5)

Real-World Applications

• Construction: Checking if corners are square
• Navigation: Finding distances
• Design: Diagonal measurements
• Physics: Vector calculations