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

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

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

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

Answer 1

To determine whether the lines intersect, we need to find the slopes of both lines.

1. **First Line through points (3, 7) and (-2, -5)**:
The slope (m) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5}
\]

2. **Second Line through points (4, 8) and (10, -2)**:
The slope (m) is calculated as:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3}
\]

Now we have the slopes of both lines:
- Slope of the first line = \(\frac{12}{5}\)
- Slope of the second line = \(-\frac{5}{3}\)

Now, let's check if they are parallel:
- Lines are parallel if their slopes are equal.
- \(\frac{12}{5} \neq -\frac{5}{3}\), so the lines are not parallel.

Next, let's check if they are perpendicular:
- Lines are perpendicular if the product of their slopes is -1:
\[
\frac{12}{5} \cdot -\frac{5}{3} = \frac{-60}{15} = -4 \quad (\text{not } -1)
\]
Thus, they are not perpendicular.

Since the lines are not parallel and not perpendicular, they must intersect.

Therefore, the correct response is:
**Yes…these lines are not parallel.**