Question

Line c passes through points (94, -48) and (2, -7). Line d passes through points (-31, -3) and (10, 89). Are line c and line d parallel or perpendicular?

Answer 1

To determine whether lines \( c \) and \( d \) are parallel or perpendicular, we first need to find the slopes of both lines.

### Step 1: Calculate the slope of Line c
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For line \( c \) which passes through points \((94, -48)\) and \((2, -7)\):

Let \( (x_1, y_1) = (94, -48) \) and \( (x_2, y_2) = (2, -7) \).

Now, we calculate the slope:

\[
m_c = \frac{-7 - (-48)}{2 - 94} = \frac{-7 + 48}{2 - 94} = \frac{41}{-92} = -\frac{41}{92}
\]

### Step 2: Calculate the slope of Line d
For line \( d \) which passes through points \((-31, -3)\) and \((10, 89)\):

Let \( (x_1, y_1) = (-31, -3) \) and \( (x_2, y_2) = (10, 89) \).

Now, we calculate the slope:

\[
m_d = \frac{89 - (-3)}{10 - (-31)} = \frac{89 + 3}{10 + 31} = \frac{92}{41}
\]

### Step 3: Determine the relationship between the slopes
To determine if the lines are parallel or perpendicular, we analyze the slopes:

1. **Parallel**: Two lines are parallel if their slopes are equal: \( m_c = m_d \).
2. **Perpendicular**: Two lines are perpendicular if the product of their slopes is \(-1\): \( m_c \cdot m_d = -1 \).

Now we have:
- \( m_c = -\frac{41}{92} \)
- \( m_d = \frac{92}{41} \)

Next, let's find the product of the slopes:

\[
m_c \cdot m_d = \left(-\frac{41}{92}\right) \cdot \frac{92}{41} = -1
\]

### Conclusion
Since the product of the slopes is \(-1\), lines \( c \) and \( d \) are **perpendicular**.