Find the Area of Rectangle that has Two similar triangle

Question

A rectangle DEBC has triangle ABC.AB and AC intersect side DE at points F and G respectively. FG = 4, The perimeter of triangle ABC is double of the perimeter of Triangle AFG. The area of Triangle ABC = 16 sq units. What is the area of DEBC?

Answer 1

Since $\Delta ABC\sim\Delta AFG$, we obtain: $$\frac{S_{\Delta AFG}}{S_{\Delta ABC}}=\left(\frac{1}{2}\right)^2,$$ which gives $$S_{\Delta AFG}=4$$ and since $$\frac{S_{\Delta AFC}}{S_{\Delta AFG}}=\frac{AC}{AG}=2,$$ we obtain $$S_{\Delta AFC}=8.$$

Thus, $$4+8=S_{\Delta CFG}=\frac{4\cdot DC}{2},$$ which gives $$DC=6$$ and $$S_{DEBC}=6\cdot8=48.$$