First find the midpoint of $A(1,4)$ and $B(7,10)$:

Next find the slope of $\overline{AB}$:

A line perpendicular to a line with slope $1$ has slope $-1$. Now use point-slope form for the line through $M(4,7)$:

Simplify:

So the correct answer is $y=-x+11$.

Since $P$, $Q$, and $R$ are on the same line and $Q$ is between $P$ and $R$, the movement from $Q$ to $R$ must match the movement from $P$ to $Q$. From $P(2,-1)$ to $Q(8,5)$, the change in $x$ is $8-2=6$ and the change in $y$ is $5-(-1)=6$. So the vector from $P$ to $Q$ is $(6,6)$. Apply that same vector starting at $Q$: $(8+6,5+6)=(14,11)$. Therefore, the coordinates of $R$ are $(14,11)$.

A perpendicular bisector of a segment has two defining properties: it is perpendicular to the segment, and it passes through the segment's midpoint. Any point on the perpendicular bisector of a segment is equidistant from the segment's endpoints. Since $R$ lies on the perpendicular bisector of $\overline{PQ}$, it must satisfy $PR=QR$. Therefore, the statement that must be true is that point $R$ is equidistant from $P$ and $Q$.

Since the bench is placed halfway along the path, first find the total distance from the fountain to the garden. Using the distance formula between $(2,1)$ and $(10,7)$:

So the full path is 10 grid units long. Because each unit represents 100 feet, the full distance is $10\cdot 100=1000$ feet. Half of that is $\frac{1000}{2}=500$ feet. Therefore, the distance from the fountain to the bench is $500$ feet.

First find the midpoint of $\overline{AB}$:

So point $C(2,2)$ is the midpoint of the segment. Next find the slope of $\overline{AB}$:

A line perpendicular to this segment has slope $\frac{4}{3}$, and the perpendicular bisector passes through the midpoint. Since $C(2,2)$ is exactly the midpoint, it must lie on the perpendicular bisector of $\overline{AB}$. Therefore, choice D must be true.