To determine whether the lines intersect, are parallel, or are perpendicular, we can first find the slopes of both lines.
Line 1: Through points (3, 7) and (-2, -5)
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (3, 7) and (-2, -5):
\[ m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]
Line 2: Through points (4, 8) and (10, -2)
Using the points (4, 8) and (10, -2):
\[ m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3} \]
Now, we have the slopes:
- Slope of line 1: \( m_1 = \frac{12}{5} \)
- Slope of line 2: \( m_2 = -\frac{5}{3} \)
Checking for parallelism: Two lines are parallel if their slopes are equal. Here, \( \frac{12}{5} \neq -\frac{5}{3} \), so the lines are not parallel.
Checking for perpendicularity: Two lines are perpendicular if the product of their slopes is -1:
\[ m_1 \cdot m_2 = \frac{12}{5} \cdot -\frac{5}{3} = -\frac{12}{3} = -4 \]
Since the product of the slopes is not -1, the lines are not perpendicular.
Since the lines are neither parallel nor perpendicular, they must intersect at some point.
Thus, the best answer is: Yes…these lines are not parallel.