One lin passes through points A and B. Another line passes through point C and D. Classify the lines as parallel, perpendicular or neither

A(-1, -7) (-2,-4)
B(-8,6) (-9,-8)

1 answer

To classify the lines formed by the points A, B and C, D, we can start by determining the slopes of the lines.

Line AB:

Points:

  • A(-1, -7)
  • B(-2, -4)

The slope \(m\) of a line 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 points A and B:

  • \(x_1 = -1\), \(y_1 = -7\)
  • \(x_2 = -2\), \(y_2 = -4\)

Calculating the slope: \[ m_{AB} = \frac{-4 - (-7)}{-2 - (-1)} = \frac{-4 + 7}{-2 + 1} = \frac{3}{-1} = -3 \]

Line CD:

Points:

  • C(-8, 6)
  • D(-9, -8)

Now calculate the slope of line CD:

  • \(x_1 = -8\), \(y_1 = 6\)
  • \(x_2 = -9\), \(y_2 = -8\)

Calculating the slope: \[ m_{CD} = \frac{-8 - 6}{-9 - (-8)} = \frac{-14}{-1} = 14 \]

Classifying the Lines:

  1. Parallel Lines: Two lines are parallel if their slopes are equal, \(m_1 = m_2\).
  2. Perpendicular Lines: Two lines are perpendicular if the product of their slopes is \(-1\), \(m_1 \cdot m_2 = -1\).

Calculating the product of the slopes: \[ m_{AB} \cdot m_{CD} = -3 \cdot 14 = -42 \]

Conclusion:

Since the slopes are neither equal (not parallel) nor do they satisfy the perpendicular condition (product is not \(-1\)), we classify the lines as neither parallel nor perpendicular.