The problem provides four points: $P(-3, -2)$, $Q(9, 1)$, $U(3, 6)$, and $V(5, -2)$. The problem is not specified, so assuming it asks if the lines $PQ$ and $UV$ are parallel, perpendicular, or neither.

1. Problem Description

The problem provides four points: $P(-3, -2)$, $Q(9, 1)$, $U(3, 6)$, and $V(5, -2)$. The problem is not specified, so assuming it asks if the lines $PQ$ and $UV$ are parallel, perpendicular, or neither.

2. Solution Steps

First, calculate the slope of line $PQ$. The slope $m$ is given by the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

For $PQ$, we have:

$m_{PQ} = \frac{1 - (-2)}{9 - (-3)} = \frac{3}{12} = \frac{1}{4}$

Next, calculate the slope of line $UV$:

$m_{UV} = \frac{-2 - 6}{5 - 3} = \frac{-8}{2} = -4$

Now, compare the slopes.

If $m_{PQ} = m_{UV}$, the lines are parallel.

If $m_{PQ} \cdot m_{UV} = -1$, the lines are perpendicular.

In this case, $m_{PQ} = \frac{1}{4}$ and $m_{UV} = -4$.

The product of the slopes is:

$\frac{1}{4} \cdot (-4) = -1$

Thus, the lines $PQ$ and $UV$ are perpendicular.

3. Final Answer

The lines PQ and UV are perpendicular.