In a 30 -- 60 -- 90 triangle, the pattern for the legs is always this:
Shorter leg: 1
Longer leg: [tex] \sqrt{3} [/tex]
Hypotenuse: 2
(Try checking that with the Pythagorean Theorem).
With this 1 - tex] \sqrt{3} [/tex] - 2 pattern, we go to the problem. Since the longer leg is tex] 16 \sqrt{3} [/tex], the shorter leg is going to be 16. Think of the tex] \sqrt{3} [/tex] as being tex] 1 \sqrt{3} [/tex], and we multiplied it by (or scaled it up) by a factor of 16.
So the triangle's short leg is 16 in this problem.
Think of the standard 30-60-90 triangle: the shortest side is 1, the other side is sqrt(3) and the hypotenuse is 2.
We can write and solve an equation of proportions to find the length of the shorter leg of the given 30-60-90 triangle:
longer leg shorter leg
--------------- ----------------
sqrt(3) 1
-------------- = ---------
16 sqrt(3) x
Then sqrt(3)*x = 16 sqrt(3), or x = 16 (answer)
