You didnt have to use the A = absinC formula, but yeah they end up crossing out.
you just had to rearrange it so that rt A1 and rt A2 are the subjects of their own equations ie.( rt A1 = x * rt A/(x+y) ) and (rt A2 = y * rt A / (x+y) )
and then when you add both rt A1 and rt A2 they cross out...