Question

what could be the ratio of the length of the longer leg in a 30-60-90 triangle to the length of its hypotnuse

Answer 1

In a typical 30-60-90 triangle (which happens to be a right triangle, by the way), the sides, in increasing order, would be multiples of 1, sqrt(3) and 2.

Here, sqrt(3) is the longer leg (it's longer than 1).  
Thus, the ratio of the length of the longer leg to the length of the hypotenuse 

is    sqrt(3)
      ---------- 
           2