The 45-45-90 Triangle

Take any square and slice it along a diagonal. You get two identical right triangles, each with two 45 degree angles and one 90 degree angle, sitting between two legs that are the sides of the square. Since the legs are equal, the triangle is isosceles, and the angles opposite equal sides are equal too — that's where the two 45s come from. The diagonal becomes the hypotenuse, and by the Pythagorean theorem, leg² + leg² = hypotenuse², so 2(leg²) = hypotenuse², which gives hypotenuse = leg x root 2. That ratio, 1:1:root 2, never changes, no matter how big the square is, because scaling a square scales both legs and the diagonal by the same factor.

This shortcut only fires when the angle between the two equal sides is exactly 90 degrees. If you're handed an isosceles triangle with two equal sides but a vertex angle that isn't 90, root 2 does not apply — you need the law of cosines to find the third side instead, since that law handles any vertex angle and reduces to the Pythagorean theorem exactly when the angle is 90.

The 45-45-90 triangle sits next to the 30-60-90 triangle as the two special right triangles worth knowing by ratio instead of rebuilding with the Pythagorean theorem every time. Both let you skip straight to side lengths once you spot the angles, which speeds up any problem built from squares, diagonals, or isosceles right triangles.