If an equilateral triangle, having centroid at the origin has a side along the line, $x + y = 2$, then the area (in sq. units) of this triangle is 

As triangle is equilateral, O is orthocentre of the triangle, and 

$y=x$ is altitude of the triangle, then 

$\mathrm{AM}=\frac{\sqrt{3}}{2}a$

$\begin{array}{l}\Rightarrow \mathrm{OM}=\frac{1}{3}\left(\mathrm{AM}\right)\\ \Rightarrow \sqrt{2}=\frac{1}{3}\left(\frac{\sqrt{3}}{2}a\right)\\ \Rightarrow a=2\sqrt{6}\end{array}$

$\text{ Area of }\mathrm{ABC}=\frac{\sqrt{3}}{4}\left({\mathrm{a}}^2\right)=6\sqrt{3}$