A right triangle has legs of length 3 and 4. What is the length of the hypotenuse?

Pythagorean theorem: in a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse (the longest side, opposite the right angle). Here: 3² + 4² = 9 + 16 = 25, so c² = 25, and c = √25 = 5. The 3-4-5 right triangle is a famous Pythagorean triple — memorize it and its multiples (6-8-10, 9-12-15, etc.). Other common Pythagorean triples to memorize: 5-12-13, 8-15-17, 7-24-25. Note: the hypotenuse is always the longest side. The Pythagorean theorem only applies to RIGHT triangles. Common applications: finding distances, ladder problems, navigation, construction. The theorem extends to coordinate geometry as the distance formula: distance = √((x₂-x₁)² + (y₂-y₁)²), which is just the Pythagorean theorem in disguise.