Determine whether line segment $RS$ and line segment $TU$ are parallel, perpendicular, or neither. Given $R(-1,4)$, $S(-3,7)$, $T(5,-1)$, $U(6,1)$.

1. Problem Description

Determine whether line segment $RS$ and line segment $TU$ are parallel, perpendicular, or neither. Given $R(-1,4)$, $S(-3,7)$, $T(5,-1)$, $U(6,1)$.

2. Solution Steps

First, we need to find the slopes of the two line segments. The slope $m$ of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

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

For the line segment $RS$, with $R(-1,4)$ and $S(-3,7)$:

$m_{RS} = \frac{7 - 4}{-3 - (-1)} = \frac{3}{-3 + 1} = \frac{3}{-2} = -\frac{3}{2}$

For the line segment $TU$, with $T(5,-1)$ and $U(6,1)$:

$m_{TU} = \frac{1 - (-1)}{6 - 5} = \frac{1 + 1}{1} = \frac{2}{1} = 2$

Now we compare the slopes to determine the relationship between the line segments.

If the slopes are equal ($m_{RS} = m_{TU}$), the line segments are parallel.

If the product of the slopes is $-1$ ($m_{RS} \cdot m_{TU} = -1$), the line segments are perpendicular.

If neither of these conditions is met, the line segments are neither parallel nor perpendicular.

Check if the slopes are equal:

$-\frac{3}{2} \neq 2$

Check if the line segments are perpendicular:

$m_{RS} \cdot m_{TU} = (-\frac{3}{2})(2) = -3$

Since $-3 \neq -1$, the lines are not perpendicular.

3. Final Answer

Neither.