jmc

The shortest line from the point $(6,0)$ to the given line will be perpendicular to it. A line perpendicular to $y=2x-2$ will have a slope of $-1/2$. This will give it a form of $y=-\frac{1}{2}x+b$. Substituting the point $(6,0)$ that we know must lie on this line, we find: $$0=-\frac{1}{2}\cdot 6+b$$ $$3=b$$ The equation of the perpendicular line is $y=-\frac{1}{2}x+3$. Now, we can solve for the point where the two lines intersect: $$-\frac{1}{2}x+3=2x-2$$ $$5=\frac{5}{2}x$$ $$x=2$$ Plugging into either line, we find the point of intersection is $(2,2)$. The coordinate plane now looks like: The distance from the point $(6,0)$ to this point is: $$\sqrt{(6-2{)}^2+(0-2{)}^2}=\sqrt{16+4}=\boxed{2\sqrt{5}}$$