To determine if lines A and B are parallel or perpendicular, we need to find the slopes of both lines.
Step 1: Find the slope of Line A
The formula for the slope (\(m\)) given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For Line A, the points are \((4, 7)\) and \((9, 9)\):
\[ m_A = \frac{9 - 7}{9 - 4} = \frac{2}{5} \]
Step 2: Find the slope of Line B
Using the same formula for Line B, with points \((3, 10)\) and \((1, 15)\):
\[ m_B = \frac{15 - 10}{1 - 3} = \frac{5}{-2} = -\frac{5}{2} \]
Step 3: Compare the slopes
- Lines are parallel if their slopes are equal: \(m_A = m_B\).
- Lines are perpendicular if the product of their slopes is \(-1\): \(m_A \cdot m_B = -1\).
Now let's check the conditions:
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Check if they are parallel: \[ \frac{2}{5} \neq -\frac{5}{2} \] So, they are not parallel.
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Check if they are perpendicular: \[ m_A \cdot m_B = \left(\frac{2}{5}\right) \cdot \left(-\frac{5}{2}\right) = -1 \]
Since the product of the slopes is \(-1\), the lines are perpendicular.
Conclusion: Lines A and B are perpendicular.