Find the Area of Rectangle that has Two similar triangle

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?

• Is A in the interior or exterior of the rectangle. What does a rectangle "having" a triangle mean? – fleablood Jan 28 at 19:34
• If you can prove the triangles are symmetric then twice the perimeter means four times the area. – fleablood Jan 28 at 19:42
• Sir/Madam, fleablood, The A is actually gets outside of the rectangle if you draw AB and AC lines from points B and C which has to intersects side DE at point F and G. – Ghost Jan 28 at 19:45
• The two triangles has three similar angles which makes it similar triangle – Ghost Jan 28 at 19:50
• "The A is actually gets outside of the rectangle if you draw AB and AC lines from points B and C which has to intersects side DE at point F and G" Not if you extend $AB$ past $A$ and $AC$ past $A$ and they intersect $DE$ on the other side. – fleablood Jan 28 at 19:54

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 $$S_{DEBC}=6\cdot8=48.$$
• @Ghost $CD$ is an altitude to side $GF$ from $C$ of the $\Delta CFG$. I don't know to draw in the net. I drew it by the given. – Michael Rozenberg Jan 28 at 20:42
• $S_{\Delta CFG}=S_{\Delta AFG}+S_{\Delta AFC}.$ – Michael Rozenberg Jan 28 at 20:48