Lesson 9: Perpendicular Lines

Learning Objectives

By the end of this lesson, students will be able to:

  • Identify perpendicular lines from their equations
  • Determine if lines are perpendicular using gradients
  • Find equations of perpendicular lines

Perpendicular Lines

Two lines are perpendicular if they meet at a right angle (90°).

ImportantKey Property of Perpendicular Lines

If two lines are perpendicular, then \(m_1 \times m_2 = -1\)

Or equivalently: \(m_2 = -\frac{1}{m_1}\)

Example 1: Identifying Perpendicular Lines

Are the lines \(y = 2x + 1\) and \(y = -\frac{1}{2}x + 3\) perpendicular?

Solution: \[m_1 \times m_2 = 2 \times \left(-\frac{1}{2}\right) = -1\]

Yes, the lines are perpendicular.

Example 2: Finding a Perpendicular Line Equation

Find the equation of the line perpendicular to \(y = 4x - 2\) that passes through the point (8, 3).

Solution: - Original gradient: \(m_1 = 4\) - Perpendicular gradient: \(m_2 = -\frac{1}{4}\) - Using point-slope form with (8, 3): \[y - 3 = -\frac{1}{4}(x - 8)\] \[y - 3 = -\frac{1}{4}x + 2\] \[y = -\frac{1}{4}x + 5\]

Practice Exercises

  1. Determine if these pairs of lines are perpendicular or not:

    • \(y = 3x + 2\) and \(y = -\frac{1}{3}x + 7\)
    • \(y = 2x + 1\) and \(y = -\frac{1}{2}x + 4\)
    • \(y = 5x\) and \(y = -5x + 3\)

  2. Find the equation of the line perpendicular to \(y = \frac{1}{3}x - 1\) through point (6, 2).

  3. The line \(L_1\) passes through points A(1, 2) and B(5, 10). Find the equation of the line perpendicular to \(L_1\) through D(3, 4).

Worksheet 3B: Perpendicular Lines Practice

  1. Determine whether each pair of lines is perpendicular or not:
    • \(y = 2x + 5\) and \(y = -\frac{1}{2}x + 2\)
    • \(y = 3x - 1\) and \(y = 2x + 4\)
  2. Find the equation of:
    • The line perpendicular to \(y = -3x + 1\) through point (6, 4)
    • The line perpendicular to \(y = 5x + 2\) through point (2, 3)
  3. Challenge problems:
    • Line L passes through A(1, 2) and B(7, 8). Find the equation of the line perpendicular to L that passes through the midpoint of AB.
    • Points P(2, 3), Q(6, 5), and R(x, y) form a right angle at Q. If the line PQ has gradient \(\frac{1}{2}\), find the gradient of QR.

Problem-Solving Strategy

When working with perpendicular lines:

  1. Identify or calculate the gradient of the given line
  2. Find the negative reciprocal gradient
  3. Use the point-slope form with the given point
  4. Simplify to the required form

Real-World Applications

  • Architecture: Ensuring walls meet at right angles
  • Navigation: Plotting perpendicular courses
  • Engineering: Designing right-angle joins

Homework

Complete the perpendicular lines practice set.

  1. Perpendicular, not perpendicular
  2. \(y = \frac{1}{3}x + 2\), \(y = -\frac{1}{5}x + \frac{13}{5}\)
  3. \(y = -x + 9\), \(m = -2\)

Previous Lesson: Lesson 8: Parallel Lines
Next Lesson: Lesson 10: Equation of a Line from Gradient and a Point