Question

Line j passes through points (6, 1) and (8, 4). Line k passes through points (1, 1) and (7, 8). Are line j and line k parallel or perpendicular?

Answer 1

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

**Step 1: Calculate the slope of line \( j \)**

Line \( j \) passes through the points \( (6, 1) \) and \( (8, 4) \).

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 \( j \):
- \( (x_1, y_1) = (6, 1) \)
- \( (x_2, y_2) = (8, 4) \)

Plugging in the coordinates:

\[
m_j = \frac{4 - 1}{8 - 6} = \frac{3}{2}
\]

**Step 2: Calculate the slope of line \( k \)**

Line \( k \) passes through the points \( (1, 1) \) and \( (7, 8) \).

Using the same slope formula:

For line \( k \):
- \( (x_1, y_1) = (1, 1) \)
- \( (x_2, y_2) = (7, 8) \)

Plugging in the coordinates:

\[
m_k = \frac{8 - 1}{7 - 1} = \frac{7}{6}
\]

**Step 3: Compare the slopes**

- Slope of line \( j \) (\( m_j \)) = \( \frac{3}{2} \)
- Slope of line \( k \) (\( m_k \)) = \( \frac{7}{6} \)

Two lines are parallel if their slopes are equal. They are perpendicular if the product of their slopes equals \(-1\).

**Step 4: Check if lines are parallel or perpendicular**

Calculate the product of the slopes:

\[
m_j \cdot m_k = \left(\frac{3}{2}\right) \cdot \left(\frac{7}{6}\right) = \frac{21}{12} = \frac{7}{4}
\]

Since \( \frac{7}{4} \) is not equal to \(-1\) and the lines have different slopes, lines \( j \) and \( k \) are neither parallel nor perpendicular.

**Conclusion:** Lines \( j \) and \( k \) are neither parallel nor perpendicular.