Which Triangle Is It?

You can tell the two special right triangles apart by their angles or by the ratio of their sides. A 45-45-90 triangle has two equal legs, and its hypotenuse is a leg times √(2). A 30-60-90 triangle has sides in the ratio 1 : √3 : 2, with the shortest side facing the 30° angle. A root in one side length is not enough on its own to tell you which triangle it is.

In a 30-60-90 triangle, find the short leg first. It faces the 30° angle, and the other two sides are worked out from it. If you are given the long leg, divide it by √(3) to get the short leg. If you are given the hypotenuse, halve it. In a 45-45-90 triangle the two legs are equal, so there is no short leg to find.