To determine if the two lines intersect, we first need to calculate the slopes of both lines.
- Calculate the slope of the line through (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): \[ m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]
- Calculate the slope of the line through (4, 8) and (10, -2):
Using the points (4, 8) and (10, -2): \[ m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3} \]
- Determine if the lines are parallel or perpendicular:
- Two lines are parallel if their slopes are equal, which they are not in this case.
- Two lines are perpendicular if the product of their slopes equals -1: \[ m_1 \cdot m_2 = \frac{12}{5} \cdot \left(-\frac{5}{3}\right) = -\frac{12}{3} = -4 \quad (\text{not } -1) \]
Since the slopes are neither equal nor do they multiply to -1, the lines are neither parallel nor perpendicular, meaning they must intersect at some point.
Based on this analysis, the correct response is: Yes…these lines are not parallel.