The formula for the midpoint $M$ of a line segment with endpoints $P(x_1, y_1)$ and $Q(x_2, y_2)$ is:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Given the endpoints $P(8, 5)$ and $Q(5, -7)$, we can substitute the coordinates into the midpoint formula:

$M = \left( \frac{8 + 5}{2}, \frac{5 + (-7)}{2} \right)$

$M = \left( \frac{13}{2}, \frac{-2}{2} \right)$

$M = (6.5, -1)$

Thus, the midpoint $M$ of the line segment is $(6.5, -1)$.

Would you like more details on how midpoint formulas work or have any further questions?

Here are 5 related questions to explore:

1. How do you calculate the distance between two points on a coordinate plane?
2. What is the slope of the line passing through the points $P(8, 5)$ and $Q(5, -7)$?
3. How do you find the equation of the line that passes through two points?
4. What happens if both endpoints of the segment have the same x-coordinates?
5. Can the midpoint formula be extended to three dimensions for 3D geometry?

Tip: Remember, the midpoint is always the average of the coordinates of the endpoints!