smc

We imagine this problem on a coordinate plane and let Alice's starting position be the origin. We see that she will travel along two edges and then go halfway along a third. Therefore, her new $x$-coordinate will be $1+2+\frac{1}{2}=\frac{7}{2}$ because she travels along a distance of $2\cdot \frac{1}{2}=1$ km because of the side relationships of an equilateral triangle, then $2$ km because the line is parallel to the $x$-axis, and the remaining distance is $\frac{1}{2}$ km because she went halfway along and because of the logic for the first part of her route. For her $y$-coordinate, we can use similar logic to find that the coordinate is $\sqrt{3}+0-\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}$. Therefore, her distance is $$\sqrt{{\left(\frac{7}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^2}=\sqrt{\frac{49}{4}+\frac{3}{4}}=\sqrt{\frac{52}{4}}=\sqrt{13},$$ giving an answer of $\boxed{\sqrt{13}}$.