What is the value of x in the diagram?

Let  h  be the height of the triangle.

From the triangle on the left:  sin(60)  =  h / 8   --->   h  = 8·sin(60)   --->   6.9282

From the triangle on the right:  sin(45)  =  6.9282 / x   --->   sin(45)·x  =  6.9282

                                --->   x  =  6.9282 / sin(45)  =  ........

x = [ sin(60º) * 8 ] / sin(45º)

in a 30-60-90 triangle the legs of the 60^o triangle has legs 4 and 4sqrt3 so we have x=sqrt(2(4sqrt3)^2) since it is an 45-45-90 triangle and we simplify to get x=sqrt(2*48) so \(x=\sqrt{96}=4\sqrt6\)