jmc

We must find the distance between each pair of points.

The distance between $(1,2)$ and $(1,12)$ is simply 10, since these two points have the same $x$-coordinate.

The distance between $(1,2)$ and $(7,10)$ is $$\sqrt{(1-7{)}^2+(2-10{)}^2}=\sqrt{36+64}=10.$$

The distance between $(7,10)$ and $(1,12)$ is $$\sqrt{(7-1{)}^2+(10-12{)}^2}=\sqrt{36+4}=2\sqrt{10}.$$

Out of 10, 10, and $2\sqrt{10}$, $2\sqrt{10}$ is the shortest value. We know this because $\sqrt{10}>\sqrt{9}$, so $\sqrt{10}>3$, so $2\sqrt{10}<(\sqrt{10}{)}^2=10$. Thus, the shortest side of the triangle has length $\boxed{2\sqrt{10}}$.