The problem asks us to find the slope of the line segment $TS$ and the slope of the line segment $QR$ given the coordinates of the vertices of the quadrilateral $QRST$: $Q(3, 2)$, $R(1, -6)$, $S(-1, -6)$, and $T(-3, 2)$.

1. Problem Description

The problem asks us to find the slope of the line segment $TS$ and the slope of the line segment $QR$ given the coordinates of the vertices of the quadrilateral $QRST$: $Q(3, 2)$, $R(1, -6)$, $S(-1, -6)$, and $T(-3, 2)$.

2. Solution Steps

The formula for the slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

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

First, we find the slope of $TS$. The coordinates of $T$ are $(-3, 2)$ and the coordinates of $S$ are $(-1, -6)$.

$m_{TS} = \frac{-6 - 2}{-1 - (-3)} = \frac{-8}{-1 + 3} = \frac{-8}{2} = -4$

Next, we find the slope of $QR$. The coordinates of $Q$ are $(3, 2)$ and the coordinates of $R$ are $(1, -6)$.

$m_{QR} = \frac{-6 - 2}{1 - 3} = \frac{-8}{-2} = 4$

3. Final Answer

Slope of $TS$: $-4$

Slope of $QR$: $4$