To solve this problem, we need to use the formula for the midpoint of a line segment. The midpoint $M$ of a line segment connecting two points $Q(x_1, y_1)$ and $R(x_2, y_2)$ is given by:

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

In this case, the midpoint $M$ has coordinates $(8, 6)$, and point $Q$ has coordinates $(9, 5)$. We are tasked with finding the coordinates of point $R(x_2, y_2)$.

Using the midpoint formula, we can set up two equations:

$\frac{9 + x_2}{2} = 8$ $\frac{5 + y_2}{2} = 6$

Solve for $x_2$:

$\frac{9 + x_2}{2} = 8 \quad \Rightarrow \quad 9 + x_2 = 16 \quad \Rightarrow \quad x_2 = 7$

Solve for $y_2$:

$\frac{5 + y_2}{2} = 6 \quad \Rightarrow \quad 5 + y_2 = 12 \quad \Rightarrow \quad y_2 = 7$

Thus, the coordinates of point $R$ are $(7, 7)$.

Would you like any further explanation or details?

Related questions:

1. How do you derive the midpoint formula?
2. What is the general equation of a line that passes through two points?
3. How can you find the distance between two points in the coordinate plane?
4. Can you explain the significance of symmetry in midpoint problems?
5. How would you find the coordinates of point $Q$ if given points $R$ and $M$?

Tip: Always verify your midpoint calculations by substituting back into the midpoint formula to ensure correctness.