To solve this problem, we need to calculate the slopes of lines $PQ$ and $RS$, then determine whether they are parallel, perpendicular, or neither by comparing the slopes.

Step 1: Slope Formula

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Step 2: Calculate the slope of line $PQ$

The coordinates of points $P$ and $Q$ are:

• $P(-9, -4)$
• $Q(-7, -1)$

Using the slope formula: $m(PQ) = \frac{-1 - (-4)}{-7 - (-9)} = \frac{-1 + 4}{-7 + 9} = \frac{3}{2}$

Step 3: Calculate the slope of line $RS$

The coordinates of points $R$ and $S$ are:

• $R(-2, 5)$
• $S(-6, -1)$

Using the slope formula: $m(RS) = \frac{-1 - 5}{-6 - (-2)} = \frac{-1 - 5}{-6 + 2} = \frac{-6}{-4} = \frac{3}{2}$

Step 4: Determine the relationship between the lines

• If the slopes are equal, the lines are parallel.
• If the product of the slopes is $-1$, the lines are perpendicular.
• Otherwise, the lines are neither.

Since $m(PQ) = \frac{3}{2}$ and $m(RS) = \frac{3}{2}$, the lines $PQ$ and $RS$ are parallel.

Final Answer:

• $m(PQ) = \frac{3}{2}$
• $m(RS) = \frac{3}{2}$
• Type of lines: Parallel

Do you have any questions or would you like more details?

Here are some related questions for further practice:

1. What would the slopes be if $PQ$ and $RS$ were perpendicular?
2. How would the result change if the slope of $PQ$ was negative?
3. Can two lines with undefined slopes be parallel?
4. What happens to the relationship of lines if their slopes are both zero?
5. What does it mean geometrically when two lines are neither parallel nor perpendicular?

Tip: When comparing slopes, remember that equal slopes indicate parallel lines, while slopes that multiply to $-1$ indicate perpendicular lines.