From diagram mid-point of $AB$ must lie on the plane
$M\left(\frac{2+6}{2},\frac{3+7}{2},\frac{4+8}{2}\right)\equiv \left(\mathrm{4,5},6\right)$
$AB$ is perpendicular to the required plane
Direction ratios of $\overrightarrow{AB}=<\mathrm{4,4},4>=<1,1,1>$
Equation of the required plane is
$1\left(x-4\right)+1\left(y-5\right)+1\left(z-6\right)=0$
$\Rightarrow x+y+z=15$
The perpendicular distance from the origin to the plane $=\left|\frac{0+0+0-15}{\sqrt{3}}\right|=5\sqrt{3}$ units