The Pythagorean theorem states that in a right triangle: **a² + b² = c²**, where c is the hypotenuse (the side opposite the right angle).
**Proof intuition:** if you draw squares on each side of a right triangle, the area of the square on the hypotenuse equals the sum of the areas of the other two squares.
**Real-world example:** You need to mount a TV antenna on your roof. The roof is 4 meters high and you need to run a cable horizontally 3 meters from the wall to the pole. How long is the cable?
c² = 4² + 3² = 16 + 9 = 25 c = √25 = **5 meters**
Other uses: GPS coordinates, computer graphics (distance between two points), construction (checking if corners are square using the 3-4-5 rule).