Question

Question
Does the line through (3, 7) and (-2, -5) intersect with the line through (4, 8) and (10, -2)?(1 point)
Responses

Yes…these lines are perpendicular
Yes…these lines are perpendicular

Yes…these lines are parallel
Yes…these lines are parallel

No…these lines are perpendicular
No…these lines are perpendicular

You cannot tell without a graph
You cannot tell without a graph

No…these lines are parallel
No…these lines are parallel

Yes…these lines are not parallel
Yes…these lines are not parallel

Answer 1

To determine if the two lines intersect, we first need to calculate the slopes of both lines.

1. **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):
\[
m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5}
\]

2. **Calculate the slope of the line through (4, 8) and (10, -2)**:

Using the points (4, 8) and (10, -2):
\[
m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3}
\]

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**.