Moment of inertia for the square plate through O, perpendicular to the plate is

$${I_{nn'}} = {1 \over {12}}m\left( {{a^2} + {a^2}} \right) = {{m{a^2}} \over 6}$$

Also, $$DO = {{DB} \over 2} = {{\sqrt 2 a} \over 2} = {a \over {\sqrt 2 }}$$

According to parallel axis theorem

$${{\mathop{\rm I}\nolimits} _{mm'}} = {I_{nn'}} + m{\left( {{a \over {\sqrt 2 }}} \right)^2}$$

$$ = {{m{a^2}} \over 6} + {{m{a^2}} \over 2} $$

$$= {{m{a^2} + 3m{a^2}} \over 6} $$

$$= {2 \over 3}m{a^2}$$