Lines and a Rectangle

We can see that $\triangle AGO \equiv \triangle EAO \equiv \triangle AOF$ are similar right-angle triangles.

We note that: $AO = \sqrt{OG^2+GA^2} = \sqrt{1^2+3^2} = \sqrt{10}$ .

$\Rightarrow OE = \sqrt{10}AO = (\sqrt{10})^2 = 10$

$\Rightarrow EF = \dfrac {\sqrt{10}}{3} OE = 10 \dfrac {\sqrt{10}}{3}$

Area of the quadrilateral $= EF \times AC = EF \times 2 \times AO = 10 \dfrac {\sqrt{10}}{3} \times 2 \times \sqrt{10} = \dfrac {200}{3} \approx \boxed{66.6}$