Question

Line p passes through points (-63, -32) and (-64, 12). Line q passes through points (39, -39) and (-5, -40). Are line p and line q parallel or perpendicular?

Answer 1

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**.