Solid Geometry Advanced – Problem 10: Find the distance from the origin to the plane

In space, let $B(2,0,0)$, $C(0,2,0)$, $D(0,0,2)$. Find the distance from the origin $O(0,0,0)$ to the plane $BCD$.

(1) The plane through the intercept points $(2,0,0)$, $(0,2,0)$, $(0,0,2)$ has equation $$\frac{x}{2}+\frac{y}{2}+\frac{z}{2}=1\quad\Longleftrightarrow\quad x+y+z=2.$$ (2) The distance from $O(0,0,0)$ to plane $x+y+z-2=0$ is $$d=\frac{|0+0+0-2|}{\sqrt{1^2+1^2+1^2}}=\frac{2}{\sqrt3}.$$

The distance from $O$ to plane $BCD$ is $\dfrac{2}{\sqrt3}$.