To determine if the two lines intersect, we first need to calculate the slopes of both lines.
Line 1 passes through the points (3, 7) and (-2, -5). The slope (m₁) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For (3, 7) and (-2, -5):
- \( x_1 = 3, y_1 = 7 \)
- \( x_2 = -2, y_2 = -5 \)
Calculating the slope:
\[ m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]
Line 2 passes through the points (4, 8) and (10, -2). Calculating the slope (m₂) as follows:
For (4, 8) and (10, -2):
- \( x_1 = 4, y_1 = 8 \)
- \( x_2 = 10, y_2 = -2 \)
Calculating the slope:
\[ m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3} \]
Now we have the slopes:
- Slope of Line 1 (m₁) = \( \frac{12}{5} \)
- Slope of Line 2 (m₂) = \( -\frac{5}{3} \)
Since the slopes are not equal, the lines are not parallel.
To check for perpendicularity, we need to see if the product of the slopes is -1:
\[ m_1 \times m_2 = \left(\frac{12}{5}\right) \times \left(-\frac{5}{3}\right) = -\frac{12}{3} = -4 \]
Since the product of the slopes is not -1, the lines are not perpendicular.
Therefore, the lines intersect and they are not parallel.
The correct answer is: Yes…these lines are not parallel.