Key Concepts and Formulas

• Midpoint Formula: The midpoint $M$ of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.
• Slope Formula: The slope $m$ of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• Parallel Lines: Parallel lines have equal slopes.
• Point-Slope Form: The equation of a line passing through $(x_0, y_0)$ with slope $m$ is $y - y_0 = m(x - x_0)$.

Step-by-Step Solution

1. Identify the vertices and the median

We are given the vertices $P(2, 2)$, $Q(6, -1)$, and $R(7, 3)$. $PS$ is the median, meaning $S$ is the midpoint of $QR$.

2. Calculate the coordinates of point S

We use the midpoint formula to find the coordinates of $S$, the midpoint of $QR$. $S = \left( \frac{6+7}{2}, \frac{-1+3}{2} \right) = \left( \frac{13}{2}, \frac{2}{2} \right) = \left( \frac{13}{2}, 1 \right)$ Thus, $S = \left( \frac{13}{2}, 1 \right)$.

3. Calculate the slope of median PS

We use the slope formula with points $P(2, 2)$ and $S\left( \frac{13}{2}, 1 \right)$ to find the slope of $PS$. $m_{PS} = \frac{1 - 2}{\frac{13}{2} - 2} = \frac{-1}{\frac{13 - 4}{2}} = \frac{-1}{\frac{9}{2}} = -\frac{2}{9}$ The slope of $PS$ is $-\frac{2}{9}$.

4. Determine the slope of the required line

The line we are looking for is parallel to $PS$, so it has the same slope. $m_{req} = m_{PS} = -\frac{2}{9}$

5. Find the equation of the required line

We use the point-slope form of a line with the point $(1, -1)$ and slope $-\frac{2}{9}$. $y - (-1) = -\frac{2}{9}(x - 1)$ $y + 1 = -\frac{2}{9}(x - 1)$ Multiply both sides by 9 to eliminate the fraction: $9(y + 1) = -2(x - 1)$ $9y + 9 = -2x + 2$ Rearrange to get the equation in the form $Ax + By + C = 0$: $2x + 9y + 9 - 2 = 0$ $2x + 9y + 7 = 0$

Common Mistakes & Tips

• Carefully handle negative signs, especially when using the slope and midpoint formulas. Double-check each calculation.
• Remember that parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.
• When rearranging the equation into the standard form, ensure all terms are moved correctly, and the signs are handled appropriately.

Summary

We found the midpoint of $QR$ to determine the coordinates of $S$. Then, we calculated the slope of the median $PS$. Since the required line is parallel to $PS$, it has the same slope. Finally, using the point-slope form with the given point $(1, -1)$ and the calculated slope, we derived the equation of the line. The equation of the line is $2x + 9y + 7 = 0$.

The final answer is \boxed{2x + 9y + 7 = 0}, which corresponds to option (D).