In a 30-60-90 triangle, the ratio of the lengths of the two legs is 1:√3. This can be proven using the Pythagorean theorem.
The hypotenuse of a 30-60-90 triangle is twice the length of the shorter leg. So, if the shorter leg is x, then the hypotenuse is 2x.
The longer leg is √3 times the length of the shorter leg. So, if the shorter leg is x, then the longer leg is √3x.
Therefore, the ratio of the lengths of the two legs is x/√3x = 1/√3.