Check the image and tell that what is the area of DEF?

Question

In the below figure , a square ABCD of area 225, E is the intersection of BD and the semicircle of diameter AD, and CF is tangent to the semicircle. Then, what is the area of DEF?

Answer 1

S1 = S/10 = 225/10 = 22.5

How

Let FD, OC meet at X

Then AFD, FXC,DXC are all congruent triangles and since each are similar to Tr.ODC, OC = sqrt5 a/ 2 where a = side of the square

Hence XD = a^2/4 /((sqrt5 a/2)/2) = a/sqrt5 = AF

S(AFD) = S(ODC) AF^2/OD ^2 = a^2/5

S(FCD) = 2a^2/5

So d the altitude from F in Tr. AFD = a^/5/(a/2) = 2a/5

So S(BFC) = (1/2)(3a/5)a = 3a^2/10

Hence S(BFD) = 2a^2/5+3a^2/10-a^2/2 = a^2/5

Therefore S(FED) = S(BFD)/2 = a^2/10 = S/10