Answer:
There are two ways to do this problem algebraically or trigonometrically.
Algebraically/geometrically
The ratios of the sides of a 30/60/90 tri. are x, x√3, 2x (short leg, long leg, hyp). Therefore, if the long leg is 6 'units'. then 6 = x√3. x = 6√3.
The hyp is 2x then the hypotenuse is 2(6√3) = 12√3, rationalizing is 12√3/3 = 4√3
Using Trig.
We can use sinx = y/r = opp/hyp. The long leg of 6 is opposite 60 degrees (pi/3).
Therefore, sin(pi/3) = 6/r =
r = 6/sin(pi/3) = 6/(√3/2) = 12/√3, when you rationalize you get 12√3/3 = 4√3
