Question

Line e passes through points (75, -44) and (42, -7). Line f passes through points (43, 88) and (80, 55). Are line e and line f parallel or perpendicular?

Answer 1

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.