To determine whether the lines \( e \) and \( f \) are parallel or perpendicular, we need to find the slopes of both lines.
Step 1: Find the slope of Line \( e \).
The formula for the slope \( m \) of a line that passes through 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 \( e \) passing through points \( (75, -44) \) and \( (42, -7) \):
- Point 1: \( (x_1, y_1) = (75, -44) \)
- Point 2: \( (x_2, y_2) = (42, -7) \)
Calculating the slope \( m_e \):
\[ m_e = \frac{-7 - (-44)}{42 - 75} = \frac{-7 + 44}{42 - 75} = \frac{37}{-33} = -\frac{37}{33} \]
Step 2: Find the slope of Line \( f \).
Line \( f \) passes through points \( (43, 88) \) and \( (80, 55) \):
- Point 1: \( (x_1, y_1) = (43, 88) \)
- Point 2: \( (x_2, y_2) = (80, 55) \)
Calculating the slope \( m_f \):
\[ m_f = \frac{55 - 88}{80 - 43} = \frac{-33}{37} \]
Step 3: Determine if lines are parallel or perpendicular.
Two lines are parallel if they have the same slope. Two lines are perpendicular if the product of their slopes is \( -1 \).
- Slope of Line \( e \): \( m_e = -\frac{37}{33} \)
- Slope of Line \( f \): \( m_f = -\frac{33}{37} \)
Calculating the product of the slopes:
\[ m_e \times m_f = \left(-\frac{37}{33}\right) \times \left(-\frac{33}{37}\right) = \frac{37 \times 33}{33 \times 37} = 1 \]
Since the product of the slopes \( m_e \times m_f = 1 \), the lines are neither parallel nor perpendicular but rather they are expressed as lines with a positive product slope.
Thus, Lines \( e \) and \( f \) are not parallel and not perpendicular.