To find the slopes of the lines that pass through the given points, we use the formula for the slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Line 1: Points (4, -3) and (6, 2)
Let \((x_1, y_1) = (4, -3)\) and \((x_2, y_2) = (6, 2)\).
\[ m_1 = \frac{2 - (-3)}{6 - 4} = \frac{2 + 3}{6 - 4} = \frac{5}{2} \]
So, the slope of line 1 is \( m_1 = \frac{5}{2} \).
Line 2: Points (3, -2) and (13, 2)
Let \((x_1, y_1) = (3, -2)\) and \((x_2, y_2) = (13, 2)\).
\[ m_2 = \frac{2 - (-2)}{13 - 3} = \frac{2 + 2}{13 - 3} = \frac{4}{10} = \frac{2}{5} \]
So, the slope of line 2 is \( m_2 = \frac{2}{5} \).
Determine the relationship between the lines
- Parallel lines have the same slope \( m_1 = m_2 \).
- Perpendicular lines have slopes that are negative reciprocals of each other, i.e., \( m_1 \cdot m_2 = -1 \).
Now, we check if the lines are parallel or perpendicular:
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Slopes:
- Slope of line 1: \( m_1 = \frac{5}{2} \)
- Slope of line 2: \( m_2 = \frac{2}{5} \)
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Check for parallel:
- \( m_1 \neq m_2 \) (since \( \frac{5}{2} \neq \frac{2}{5} \))
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Check for perpendicular:
- Multiply the slopes: \[ m_1 \cdot m_2 = \frac{5}{2} \cdot \frac{2}{5} = \frac{10}{10} = 1 \quad \text{(not } -1\text{)} \]
Conclusion
The slopes are:
- Slope of line 1: \(\frac{5}{2}\)
- Slope of line 2: \(\frac{2}{5}\)
The two lines are neither parallel nor perpendicular.
So, filling in the response areas:
- The slope of line 1 is \(\frac{5}{2}\).
- The slope of line 2 is \(\frac{2}{5}\).
- These lines are neither parallel nor perpendicular.