To determine if the two lines intersect, we can find their slopes. If the slopes are different, the lines will intersect unless they are collinear. If the slopes are the same, the lines are parallel. If the product of the slopes is -1, the lines are perpendicular.
Step 1: Find the slope of the line through the points (3, 7) and (-2, -5).
The formula for 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 \((-2, -5)\):
- \( (x_1, y_1) = (3, 7) \)
- \( (x_2, y_2) = (-2, -5) \)
Substituting into the slope formula:
\[
m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5}
\]
Step 2: Find the slope of the line through the points (4, 8) and (10, -2).
Using the points \((4, 8)\) and \((10, -2)\):
- \( (x_1, y_1) = (4, 8) \)
- \( (x_2, y_2) = (10, -2) \)
Substituting into the slope formula:
\[
m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = \frac{-5}{3}
\]
Step 3: Compare the slopes
- The slope of the first line \( m_1 = \frac{12}{5} \)
- The slope of the second line \( m_2 = \frac{-5}{3} \)
Step 4: Determine the relationship between the lines
-
Are the lines parallel?
Since \( \frac{12}{5} \) is not equal to \( \frac{-5}{3} \), the lines are not parallel.
-
Are the lines perpendicular?
To check if the lines are perpendicular, we need to determine if the product of their slopes is -1:
\[
m_1 \cdot m_2 = \frac{12}{5} \cdot \frac{-5}{3} = \frac{12 \cdot -5}{5 \cdot 3} = -\frac{12}{3} = -4
\]
The product is not -1, so the lines are not perpendicular.
Conclusion
Since the slopes are different, the two lines intersect. Therefore, the appropriate choice is:
Yes…these lines are not parallel.
