smc

Draw height $CH$ (Perpendicular line from point C to line AD). We have that $BH=1$. By the Pythagorean Theorem, $CH=\sqrt{48}$. Since $CD=8$, $HD=\sqrt{8^2-48}=\sqrt{16}=4$, and $BD=HD-1$, so $BD=\boxed{3}$.