What is the area of triangle AFE?

Question

If ED = 23 , and the value of the side of the square ABCD is a multiple of 11, what is the area of the red triangle AFE?! Find the very shortest way to solve this puzzle and use only basic geometry, trigonometry is not allowed.

Answer 1

Let $AB = 11x$. Triangles EDF and EAB are similar, so:

$\dfrac{ED}{EA} = \dfrac{DF}{AB}$

$\dfrac{23}{23 + 11x} = \dfrac{DF}{11x}$

$DF = \dfrac{253x}{23 + 11x}$

The area of $\triangle AFE$ is thus

$$\dfrac{1}{2} \cdot EA \cdot DF = \dfrac{1}{2} \cdot (23 + 11x) \cdot \dfrac{253x}{23 + 11x} = \dfrac{253}{2}x$$

Answer 2

let us consider one simple situation,suppose AB=33; you can check that 33 is multiple of 11,33/11=3,and also we know that DF/FC=1/2.if we denote DF by x,then FC=2*x so x+2*x=33, 3*x=33 x=11;(sorry in first coment instead of DC should be FC) so DF=11; length of AE=AD+DE or 33+23=56,so are of AEF=1/2*DF*AE`=1/2*56*11=28*11=308