+195 In the diagram,\( \triangle BDF\) and \(\triangle ECF \) have the same area. If \(DB=2,BA=3,\) find the length of \(\overline{EC}.\)
+79 We know that $\triangle BDF$ and $\triangle ECF$ have the same area, and adding the area of quadrilateral $ABFE$ to each of them shows us that$$\text{(area of $\triangle ADE$)}=\text{(area of $\triangle ABC$)}.$$Using the area formula for triangles, we can turn the last equation into $$\dfrac{1}{2}\cdot AD\cdot AE = \dfrac{1}{2}\cdot AB\cdot AC.$$Substituting in the given lengths, we get $$\dfrac{1}{2}\cdot 5\cdot 4 = \dfrac{1}{2}\cdot 3\cdot AC,$$so $AC=\dfrac{20}{3}$. We compute the length we are interested in as $EC=AC-AE=\dfrac{20}{3}-4=\boxed{\dfrac{8}{3}}$.
Your answer is based on the assumption that triangles ABC and ADE are not congruent.
Based on the given information, one can argue that these triangles are in fact congruent, and the answer will be a lot different.
btw, where did the number 4 come from? Why not number 3?!? 
