Pythagorean relation
If ∠C is right, then a² + b² = c², where c is the hypotenuse.
The converse is also true: if a² + b² = c², the angle opposite c is right.
Special right triangles
45°–45°–90°: legs equal; hypotenuse = leg × √2.
30°–60°–90°: short leg opposite 30°; hypotenuse = 2 × short; long leg = short × √3.
Altitude to the hypotenuse
The altitude to the hypotenuse creates two smaller right triangles, each similar to the original and to each other.
Geometric mean: each leg is the geometric mean of the hypotenuse and the adjacent segment.