find the length of diagonal AC.

Question

ABCD is a square.F is mid-point of AB and BE is one-third of BC.if the area of triangle FBE is 49 sq.cm,find the length of diagonal AC.

Answer 1

If the square has side length a, then the area of triangle FBE is 1/2*a/2*a/3=49 sq cm (half base FB*height BE).
So a^2/12=49 (FB=a/2 and BE=a/3). So a^2=12*49 and a=14sqrt(3).
(sqrt(12)=sqrt(4*3)=2sqrt(3).) Diagonal AC^2=2a^2 (Pythagoras), so AC=a*sqrt(2)=14sqrt(6)=34.29cm approx.