First find the slope of the line through $(4,7)$ and $(11,0)$:

So the equation of the line has the form $y=-x+b$. Substitute the point $(4,7)$:

That means the line crosses the $y$-axis at $(0,11)$. The distance from the origin $(0,0)$ to $(0,11)$ is

So the correct answer is $11$.

The midpoint of $\overline{AB}$ is found by averaging the coordinates of the endpoints:

The distance between $A(a,b)$ and $B(c,d)$ is

Since $(a-c{)}^2+(b-d{)}^2=100$, the length of $\overline{AB}$ is

Therefore, the statement that must be true is that the midpoint is $(3,-2)$ and the length is $10$.

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.

Use the midpoint formula. For endpoints $A(x_1,y_1)$ and $B(x_2,y_2)$, the midpoint is

Here, the $x$-coordinate of the midpoint is 2, so

Multiply both sides by 2:

Then add 3 to both sides:

Therefore, the correct answer is $7$.

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

For the points $(2,-1)$ and $(8,5)$, the horizontal change is $8-2=6$ and the vertical change is $5-(-1)=6$. So

Therefore, the distance between the stations is $6\sqrt{2}$.