Special triangles and ratio of areas help?
I've been lurking around here for a round a week and found you guys are really helpful. Just today, I've been hit with two hard problems (to me, at least. The class geniuses had no problems whatsoever with these) I was sent home and was told to bring in the answer on Monday.
So I've attached the problems, and there are some scribbles on them. Please take a look at them and see if you can help.
So for the first one, since the diagonals add up to 28, and I discovered it consisted of 4 30:60:90 triangles, I took one of the 4 small triangles. The side facing 30 became x, and the side facing 60 is x(sqrt 3).
So then x + x (sqrt 3) = 14
Now, I use pythagorean;s theorem
x^2 + ( 3 )x^2 = c^2, where C is an outer side.
Now I get 2 equations - 4x^2= c^2
and x (1 + (sqrt 3)) = 14
so I get x = 14/ (1 + (sqrt 3))
I plug that in, I get c^2= 4x^2
c^2 = (4 x 196) / (4 + (sqrt 3))
c^2 = 784 / (4 + (sqrt 3))
Now I'm stuck. Can anyone help me?