To determine whether the two lines intersect, we need to find the slopes of each line.
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Find the slope of the line through (3, 7) and (-2, -5):
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 7}{-2 - 3} = \frac{-12}{-5} = \frac{12}{5} \]
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Find the slope of the line through (4, 8) and (10, -2):
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 8}{10 - 4} = \frac{-10}{6} = \frac{-5}{3} \]
Now we compare the slopes:
- The slope of the first line: \( \frac{12}{5} \)
- The slope of the second line: \( \frac{-5}{3} \)
Since the slopes are not equal, the lines are not parallel.
Next, we can check if the lines are perpendicular by seeing if the product of their slopes equals -1:
\[ m_1 \cdot m_2 = \frac{12}{5} \cdot \frac{-5}{3} = \frac{12 \cdot -5}{5 \cdot 3} = \frac{-60}{15} = -4 \]
Since the product of the slopes is not -1, the lines are not perpendicular.
Since the lines are neither parallel nor perpendicular, they do intersect at some point.
Therefore, the answer is:
Yes... these lines are not parallel.