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)

1 answer

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.