The Pythagorean Theorem

A right triangle has one 90 degree angle. The two sides touching that angle are the legs, and the side opposite it, the longest side, is the hypotenuse. The theorem says a² + b² = c², where a and b are the legs and c is the hypotenuse. This isn't a rule someone made up. Build a square on each leg and a square on the hypotenuse, and the areas of the two smaller squares always add up exactly to the area of the big one. That's why 3, 4, 5 works every time: 3² + 4² = 9 + 16 = 25 = 5^2. Any triangle whose sides fit that pattern must have a right angle, and any right triangle's sides must fit it.

To use it, identify which side is missing. If you're missing the hypotenuse, square both legs, add, then take the square root. If you're missing a leg, square the hypotenuse, subtract the square of the other leg, then take the square root. The legs always get squared and added; the hypotenuse is the one you isolate by square-rooting at the end.

This theorem is the engine behind distance problems, diagonal lengths of rectangles and boxes, and even the distance formula in coordinate geometry, which is just the theorem in disguise. Once a problem gives you two sides of a right triangle, or lets you build one from a diagram, a² + b² = c² is almost always the fastest way in.