Geometry problem only needed hint one more

Question

I just want hint how to use the congruent condition..which involves an equation in X I guess

And I obtained one equation $$y^2=x (16+x)$$

Answer 1

The condition $AC\simeq BC$ tells you that $\angle ABF=\angle BCD$. So, $ABF$ and $DCF$ are similar triangles. You get another equation from here.

Answer 2

Since $$\measuredangle BCD=\measuredangle CAB=\measuredangle B,$$ we obtain $$AB||CD$$ and from here we have the following system: $$y^2=x(x+16)$$ and $$\frac{y}{3}=\frac{x+10}{6}.$$ I got $$x=2.$$