To find the midpoint of segment $PR$, average the $x$-coordinates and average the $y$-coordinates of points $P(-4,2)$ and $R(6,4)$.

Midpoint $=\left(\frac{-4+6}{2},\frac{2+4}{2}\right)=\left(\frac{2}{2},\frac{6}{2}\right)=(1,3)$.

So the midpoint of segment $PR$ is $(1,3)$.

A point on the perpendicular bisector of a segment is equidistant from the segment’s two endpoints. Since $R$ is on the perpendicular bisector of $\overline{PQ}$, it must satisfy $RP=RQ$.

You can verify the geometry of $\overline{PQ}$ if needed: the midpoint of $P(-4,6)$ and $Q(8,-2)$ is $\left(\frac{-4+8}{2},\frac{6+(-2)}{2}\right)=(2,2)$. The perpendicular bisector is the line that passes through $(2,2)$ and is perpendicular to $\overline{PQ}$. Any point on that line is the same distance from $P$ and $Q$. Therefore, the statement that must be true is that the distance from $R$ to $P$ equals the distance from $R$ to $Q$.

To find the point halfway between $(2,1)$ and $(10,7)$, use the midpoint formula:

Substitute the coordinates:

So the water station should be placed at $(6,4)$.

Since the checkpoint $(5,3)$ is the midpoint of segment $WS$, the store's coordinates can be found from the midpoint formula. Let the store be $(x,y)$. Then

So,

and

Therefore, the store is at $(8,7)$. Now find the slope of the line through $(2,-1)$ and $(8,7)$:

Use point-slope or slope-intercept form with point $(2,-1)$:

Simplify:

So the equation of the line is $y=\frac{4}{3}x-\frac{11}{3}$.

Use the distance formula for points $(x_1,y_1)$ and $(x_2,y_2)$:

Here, the points are $(-4,6)$ and $(8,-2)$. Substituting gives

This matches choice A. Notice that the distance formula uses the differences in the $x$-coordinates and the differences in the $y$-coordinates, then squares those differences.