To determine if the lines through the given points intersect, we need to find the slopes of both lines and see if they are equal (which would mean the lines are parallel) or not.
Line 1: Through points (3, 7) and (-2, -5)
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For points (3, 7) and (-2, -5):
\[ m_1 = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]
Line 2: Through points (4, 8) and (10, -2)
Using the same formula:
\[ m_2 = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = -\frac{5}{3} \]
Now that we have the slopes of both lines:
- Slope of Line 1: \( \frac{12}{5} \)
- Slope of Line 2: \( -\frac{5}{3} \)
Since the slopes are different, the lines are not parallel.
Now, to determine if they are perpendicular, we check if the product of the slopes is -1:
\[ m_1 \cdot m_2 = \left( \frac{12}{5} \right) \cdot \left( -\frac{5}{3} \right) = -\frac{60}{15} = -4 \]
Since the product is not -1, the lines are not perpendicular either.
From the analysis, the lines are neither parallel nor perpendicular, and since they are not parallel, they must intersect.
Thus, the correct response is:
Yes…these lines are not parallel.
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