Question 269934
find the exact lengths of the legs of a 30-60-90 degree triangle if the hypotneuse is 12 cm

In a 30-60-90 right triangle the side opposite the 30 degree angle is half the hypotenuse, in this case 12/2 = 6cm. Let x be the remaining side. 

Using the Pythagorean Theorem we have then:

12^2 = 6^2 + x^2

x^2 = 144 - 36
x^2 = 108
x = sqrt(108) = sqrt(36*3) = sqrt(36)*sqrt(3) 
x = 6*sqrt(3)