8. Given that $A B C D$ is a square with side length 4, $E, F$ are the midpoints of $A B, A D$ respectively, $G C \perp$ plane $A B C D$, and $G C=2$. Find the distance from point $B$ to plane $E F G$.
8. With the square $ABCD$ as the base and $GC$ as the edge, construct the rectangular prism $ABCD-A'B'G D'$. Since $BD // $ plane $EFG$, the distance from $B$ to plane $EFG$ is equal to the distance from line $BD$ to plane $EFG$, which is the distance from the center $O$ of $ABCD$ to plane $EFG$.

Draw $OK \perp GH$ at $K$ (where $H$ is the intersection of $EF$ and $AC$), then $OK \perp$ plane $EFG$, and segment $OK$ is the distance from point $O$ to plane $EFG$. Given $GC=2, CH=3\sqrt{2}, OH=\sqrt{2}$, then $GH=\sqrt{GC^2 + CH^2} = \sqrt{22}$. Also, $\frac{OK}{GC} = \frac{OH}{GH}$, hence
$$
OK = \frac{OH \cdot GC}{GH} = \frac{2\sqrt{11}}{11}.
$$