In more detail
The hypotenuse c is the side opposite the right angle and always the longest. Sides 3 and 4 give a hypotenuse of 5; other whole-number triples are 5-12-13 and 8-15-17, and any multiple of a triple (6-8-10) works too. The converse also holds: if a² + b² = c², the triangle has a right angle. In three dimensions the same idea gives √(x² + y² + z²), and in a plane it gives the distance between two points: √(Δx² + Δy²).
Why it matters
Builders check corners are square by measuring 3, 4 and 5 units (or 6, 8 and 10) along the sides. Screen sizes, ladder heights, diagonal braces, map distances and vector lengths all use it. It applies only to right triangles; for any other triangle use the law of cosines from Trigonometry.
Example
5 m ladder, foot 1.5 m from the wall
height = √(5² − 1.5²) = √22.75 = 4.77 m
55-inch 16:9 TV (diagonal 55)
diagonal of a 16 × 9 box = √(16² + 9²) = √337
width = 55 × 16 / √337 = 47.94 in
height = 55 × 9 / √337 = 26.96 in
distance from (1, 2) to (4, 6) = √(3² + 4²) = 5
distance from (0, 0, 0) to (1, 2, 2) = √(1 + 4 + 4) = 3