To determine whether the lines \( p \) and \( q \) are parallel, perpendicular, or neither, we need to calculate the slopes of both lines.
The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For line \( p \), which passes through points \((-63, -32)\) and \((-64, 12)\):
Let \((x_1, y_1) = (-63, -32)\) and \((x_2, y_2) = (-64, 12)\).
Calculating the slope:
\[ m_p = \frac{12 - (-32)}{-64 - (-63)} = \frac{12 + 32}{-64 + 63} = \frac{44}{-1} = -44 \]
For line \( q \), which passes through points \((39, -39)\) and \((-5, -40)\):
Let \((x_1, y_1) = (39, -39)\) and \((x_2, y_2) = (-5, -40)\).
Calculating the slope:
\[ m_q = \frac{-40 - (-39)}{-5 - 39} = \frac{-40 + 39}{-5 - 39} = \frac{-1}{-44} = \frac{1}{44} \]
Now we have the slopes:
- \( m_p = -44 \)
- \( m_q = \frac{1}{44} \)
Checking for Parallelism:
Two lines are parallel if their slopes are equal. Here, \( -44 \) is not equal to \( \frac{1}{44} \), so the lines are not parallel.
Checking for Perpendicularity:
Two lines are perpendicular if the product of their slopes equals \(-1\):
\[ m_p \cdot m_q = -44 \cdot \frac{1}{44} = -1 \]
Since this holds true (\(-1\)), the lines are perpendicular.
Conclusion:
Line \( p \) and line \( q \) are perpendicular.