Question

On a coordinate plane, 3 lines are shown. Line m has points (negative 4, 3) and (0, negative 4). Line n has points (1, 2) and (3, negative 2). Line k has points (negative 3, negative 3) and (4, 1).
Use the diagram to answer the questions.

Is line m parallel to line n? Explain.


Is line m perpendicular to line k? Explain.

Answer 1

To determine whether line \( m \) is parallel to line \( n \) and whether line \( m \) is perpendicular to line \( k \), we need to calculate the slopes of each line based on the points provided.

### Finding the slopes

1. **Slope of line \( m \)**:
- The points are \( (-4, 3) \) and \( (0, -4) \).
- Slope formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
- Substituting the points into the formula:
\[
m = \frac{-4 - 3}{0 - (-4)} = \frac{-7}{4}
\]

2. **Slope of line \( n \)**:
- The points are \( (1, 2) \) and \( (3, -2) \).
- Substituting the points:
\[
n = \frac{-2 - 2}{3 - 1} = \frac{-4}{2} = -2
\]

3. **Slope of line \( k \)**:
- The points are \( (-3, -3) \) and \( (4, 1) \).
- Substituting the points:
\[
k = \frac{1 - (-3)}{4 - (-3)} = \frac{1 + 3}{4 + 3} = \frac{4}{7}
\]

### Comparing the slopes

1. **Is line \( m \) parallel to line \( n \)?**
- Lines are parallel if they have the same slope.
- Slope of line \( m \): \( -\frac{7}{4} \)
- Slope of line \( n \): \( -2 = -\frac{4}{2} \)
- Since \( -\frac{7}{4} \neq -2 \), line \( m \) is **not parallel** to line \( n \).

2. **Is line \( m \) perpendicular to line \( k \)?**
- Lines are perpendicular if the product of their slopes equals \( -1 \).
- Slope of line \( m \): \( -\frac{7}{4} \)
- Slope of line \( k \): \( \frac{4}{7} \)
- Calculate the product of the slopes:
\[
-\frac{7}{4} \cdot \frac{4}{7} = -1
\]
- Since the product of the slopes is \( -1 \), line \( m \) is **perpendicular** to line \( k \).

### Summary:
- Line \( m \) is **not parallel** to line \( n \).
- Line \( m \) **is perpendicular** to line \( k \).