Question

Are they perpendicular,parallel or neither.
One line passes through point A (0,-5) and b (-3,4) and another line passed through c (6,2) and d (-3,-1)

Answer 1

To determine if the two lines are perpendicular, parallel, or neither, we need to find the slopes of both lines.

### Line 1 (Points A and B)
Points A and B are:
- A(0, -5)
- B(-3, 4)

The slope \( m_1 \) of line AB can be calculated using the formula:
\[
m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-5)}{-3 - 0} = \frac{4 + 5}{-3} = \frac{9}{-3} = -3
\]

### Line 2 (Points C and D)
Points C and D are:
- C(6, 2)
- D(-3, -1)

The slope \( m_2 \) of line CD can be calculated as:
\[
m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 2}{-3 - 6} = \frac{-3}{-9} = \frac{1}{3}
\]

### Comparing Slopes
- The slope of line AB is \( m_1 = -3 \).
- The slope of line CD is \( m_2 = \frac{1}{3} \).

1. **Parallel lines** have the same slope. Since \( -3 \neq \frac{1}{3} \), the lines are **not parallel**.
2. **Perpendicular lines** have slopes that are negative reciprocals of each other. The negative reciprocal of \( m_1 \) is:
\[
-\frac{1}{m_1} = -\left(-\frac{1}{3}\right) = \frac{1}{3}
\]
Since \( m_2 = \frac{1}{3} \), the two lines are indeed **perpendicular**.

### Conclusion
The lines that pass through points A and B are **perpendicular** to the lines that pass through points C and D.