and 
Prove that
Prove that
into rectangular form.
Y by
Point 
lies on side 
of square 
and square 
is drawn outwards , as shown in the figure below. Suppose that the area of triangle 
is 
of the area of pentagon 
. What is the ratio of the areas of and ?
lies on side 
of square 
and square 
is drawn outwards , as shown in the figure below. Suppose that the area of triangle 
is 
of the area of pentagon 
. What is the ratio of the areas of and ?
and 
.
.![\[S_{EGC}=(a+x)^2+\frac{x^2}{2}-\frac{(a+x)(a+2x)}{2}-\frac{a(a+x)}{2}=\frac{x^2}{2}.\]](http://latex.artofproblemsolving.com/2/a/7/2a7271b3023b31cbdadac9161187b92f1a292986.png)
![\[S_{DEFBC}=(a+x)^2+x^2+\frac{ax}{2}.\]](http://latex.artofproblemsolving.com/c/f/0/cf022939edf4bba3c1ce126b096a1e193665deee.png)
![\[S_{DEFBC}=16S_{EGC}\Rightarrow a^2+2.5ax-6x^2=0\]](http://latex.artofproblemsolving.com/e/a/6/ea6e9085f8aac29b7c60539cc726c5be3c33a48e.png)
![\[\left(\frac{a}{x}\right)^2+2.5\frac{a}{x}-6=0,\quad \frac{a}{x}>0\Rightarrow\frac{a}{x}=\frac{3}{2}.\]](http://latex.artofproblemsolving.com/f/3/0/f3047f7e354ad1e790923424d1205422532724df.png)
Therefore,![\[\left(\dfrac{x}{a+x}\right)^2=\left(\dfrac{1}{\frac{a}{x}+1}\right)^2=\frac{4}{25}.\]](http://latex.artofproblemsolving.com/9/3/8/9388ead9972342deafa8461cdcb8961e91254ed0.png)
Hence answer is: 
.
