The 45-45-90 Triangle

Two right triangles turn up so often that their side ratios are worth knowing. The first is the 45-45-90 triangle: half a square, cut along its diagonal. Its two legs are equal, so with legs of 1 the hypotenuse is √(1^2 + 1^2) = √(2).

Scale it up and every side scales together: legs of a give a hypotenuse of a×√(2). You can spot one from a single 45° angle in a right triangle, since the third angle is then 45° too and the legs are equal.

Here the angle at R is 45° and a leg is 5, so the hypotenuse is 5×√(2). Pythagoras gives the same: 25 + 25 = 50, and √(50) = 5×√(2). On the stage pick Special right triangle.