Determine whether the lines PQ and UV are perpendicular. The coordinates of the points are given as P(1, 1), Q(9, 8), U(-6, 1), and V(2, 8).

1. Problem Description

Determine whether the lines PQ and UV are perpendicular. The coordinates of the points are given as P(1, 1), Q(9, 8), U(-6, 1), and V(2, 8).

2. Solution Steps

To determine if the lines are perpendicular, we need to find the slopes of the lines PQ and UV. Then, we check if the product of their slopes is -

1.

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

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

Slope of line PQ:

Using the coordinates of P(1, 1) and Q(9, 8), we have:

$m_{PQ} = \frac{8 - 1}{9 - 1} = \frac{7}{8}$

Slope of line UV:

Using the coordinates of U(-6, 1) and V(2, 8), we have:

$m_{UV} = \frac{8 - 1}{2 - (-6)} = \frac{7}{2 + 6} = \frac{7}{8}$

Now, we check if the product of the slopes is -1:

$m_{PQ} \cdot m_{UV} = \frac{7}{8} \cdot \frac{7}{8} = \frac{49}{64}$

Since $\frac{49}{64} \neq -1$, the lines PQ and UV are not perpendicular.

3. Final Answer

The lines PQ and UV are not perpendicular.