How To Find The Equation Of A Line Perpendicular

Finding the equation of a line perpendicular to another line is a fundamental concept in coordinate geometry. It involves understanding the relationship between the slopes of perpendicular lines and applying this knowledge to derive the equation of the new line. This comprehensive guide will walk you through the steps, provide explanations, and offer examples to ensure you grasp the concept thoroughly.

Understanding Perpendicular Lines

Two lines are perpendicular if they intersect at a right angle (90 degrees). The key characteristic of perpendicular lines lies in the relationship between their slopes.

The Slopes of Perpendicular Lines

If a line has a slope m, then a line perpendicular to it has a slope of -1/m. In other words, the slopes of perpendicular lines are negative reciprocals of each other.

• If the slope of line 1 is m₁ and the slope of line 2 is m₂, then for the lines to be perpendicular:

  m₁ * m₂ = -1

This relationship is crucial for finding the equation of a perpendicular line.

Prerequisites

Before diving into the steps, ensure you're familiar with the following concepts:

• Slope of a Line: The slope (m) of a line is a measure of its steepness and direction. It is calculated as the change in y divided by the change in x between two points on the line:

  m = (y₂ - y₁) / (x₂ - x₁)

• Equation of a Line: There are several forms of the equation of a line:

  • Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
  • Point-Slope Form: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
  • Standard Form: Ax + By = C, where A, B, and C are constants.

Steps to Find the Equation of a Perpendicular Line

Here’s a step-by-step guide to finding the equation of a line perpendicular to a given line:

Step 1: Determine the Slope of the Given Line

Identify the slope of the given line. This may be provided directly, or you may need to calculate it from two points on the line or by rearranging the equation into slope-intercept form.

Example 1:

Find the equation of a line perpendicular to the line y = 2x + 3 and passing through the point (4, -1).

• The given line is in slope-intercept form, y = mx + b. The slope of the given line is m = 2.

Example 2:

Find the equation of a line perpendicular to the line passing through points (1, 2) and (3, 6) and passing through the point (-2, 3).

• First, calculate the slope of the given line using the formula m = (y₂ - y₁) / (x₂ - x₁):

  m = (6 - 2) / (3 - 1) = 4 / 2 = 2

• The slope of the given line is m = 2.

Example 3:

Find the equation of a line perpendicular to the line 3x + 4y = 12 and passing through the point (0, 5).

• Rearrange the equation into slope-intercept form (y = mx + b):

  4y = -3x + 12

  y = (-3/4)x + 3

• The slope of the given line is m = -3/4.

Step 2: Calculate the Slope of the Perpendicular Line

Find the negative reciprocal of the slope of the given line. If the slope of the given line is m, the slope of the perpendicular line (m_perp) is -1/m.

Example 1 (Continued):

• The slope of the given line is m = 2.
• The slope of the perpendicular line is m_perp = -1/2.

Example 2 (Continued):

• The slope of the given line is m = 2.
• The slope of the perpendicular line is m_perp = -1/2.

Example 3 (Continued):

• The slope of the given line is m = -3/4.
• The slope of the perpendicular line is m_perp = -1 / (-3/4) = 4/3.

Step 3: Use the Point-Slope Form to Find the Equation

Use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the given point through which the perpendicular line passes, and m is the slope of the perpendicular line (m_perp).

Example 1 (Continued):

• We have the point (4, -1) and the slope m_perp = -1/2.

• Plug these values into the point-slope form:

  y - (-1) = (-1/2)(x - 4)

  y + 1 = (-1/2)x + 2

Example 2 (Continued):

• We have the point (-2, 3) and the slope m_perp = -1/2.

• Plug these values into the point-slope form:

  y - 3 = (-1/2)(x - (-2))

  y - 3 = (-1/2)(x + 2)

Example 3 (Continued):

• We have the point (0, 5) and the slope m_perp = 4/3.

• Plug these values into the point-slope form:

  y - 5 = (4/3)(x - 0)

  y - 5 = (4/3)x

Step 4: Simplify the Equation (Optional)

You can simplify the equation into slope-intercept form (y = mx + b) or standard form (Ax + By = C), depending on the required format.

Example 1 (Continued):

• Starting from y + 1 = (-1/2)x + 2, isolate y:

  y = (-1/2)x + 2 - 1

  y = (-1/2)x + 1

• The equation of the perpendicular line in slope-intercept form is y = (-1/2)x + 1.

Example 2 (Continued):

• Starting from y - 3 = (-1/2)(x + 2), simplify:

  y - 3 = (-1/2)x - 1

  y = (-1/2)x - 1 + 3

  y = (-1/2)x + 2

• The equation of the perpendicular line in slope-intercept form is y = (-1/2)x + 2.

Example 3 (Continued):

• Starting from y - 5 = (4/3)x, isolate y:

  y = (4/3)x + 5

• The equation of the perpendicular line in slope-intercept form is y = (4/3)x + 5.

Alternative Method: Using the Standard Form

If the given line is in standard form Ax + By = C, there's a shortcut to find the slope of the perpendicular line.

Step 1: Identify A and B

Identify the coefficients A and B in the standard form Ax + By = C.

Example:

Find the equation of a line perpendicular to 2x + 3y = 6 and passing through the point (1, -2).

• In the equation 2x + 3y = 6, A = 2 and B = 3.

Step 2: Find the Slope of the Given Line

The slope of the given line is m = -A/B.

Example (Continued):

• The slope of the given line is m = -2/3.

Step 3: Calculate the Slope of the Perpendicular Line

The slope of the perpendicular line is the negative reciprocal of the slope of the given line, which is m_perp = -1/m = B/A.

Example (Continued):

• The slope of the perpendicular line is m_perp = -1 / (-2/3) = 3/2.

Step 4: Use the Point-Slope Form and Simplify

Use the point-slope form with the calculated slope and the given point, and then simplify to the desired form.

Example (Continued):

• We have the point (1, -2) and the slope m_perp = 3/2.

• Plug these values into the point-slope form:

  y - (-2) = (3/2)(x - 1)

  y + 2 = (3/2)x - 3/2

• Isolate y to get the slope-intercept form:

  y = (3/2)x - 3/2 - 2

  y = (3/2)x - 7/2

• The equation of the perpendicular line in slope-intercept form is y = (3/2)x - 7/2.

Common Mistakes and How to Avoid Them

1. Incorrectly Calculating the Slope:

   • Mistake: Swapping x and y values when calculating the slope, or not subtracting in the correct order.
   • Solution: Double-check the formula m = (y₂ - y₁) / (x₂ - x₁) and ensure you subtract the y-values and x-values in the same order.

2. Forgetting to Take the Negative Reciprocal:

   • Mistake: Finding the reciprocal but not changing the sign.
   • Solution: Remember that the slopes of perpendicular lines are negative reciprocals. Always change the sign after finding the reciprocal.

3. Using the Wrong Point or Slope:

   • Mistake: Mixing up the given point or using the slope of the original line instead of the perpendicular line.
   • Solution: Clearly label and distinguish between the point and slope for the original line and the perpendicular line.

4. Algebra Errors:

   • Mistake: Making mistakes while simplifying the equation, such as incorrect distribution or combining like terms.
   • Solution: Take your time and double-check each step of the simplification process.

Advanced Concepts and Applications

1. Perpendicular Bisectors:

   • A perpendicular bisector is a line that is perpendicular to a line segment and passes through its midpoint. To find the equation of a perpendicular bisector:
     • Find the midpoint of the line segment using the midpoint formula: ((x₁ + x₂) / 2, (y₁ + y₂) / 2).
     • Find the slope of the line segment using the slope formula.
     • Find the negative reciprocal of the slope to get the slope of the perpendicular bisector.
     • Use the point-slope form with the midpoint and the slope of the perpendicular bisector to find the equation.

2. Applications in Geometry:

   • Perpendicular lines are used in various geometric proofs and constructions, such as finding altitudes of triangles, determining the shortest distance from a point to a line, and constructing right angles.

3. Applications in Physics and Engineering:

   • In physics, perpendicular components are used to analyze forces and motion. In engineering, perpendicularity is crucial in structural design, ensuring stability and balance.

Real-World Examples

1. Road Intersections:

   • Roads often intersect at right angles to ensure safe and efficient traffic flow. The concept of perpendicular lines is used in designing these intersections.

2. Building Construction:

   • Builders use perpendicular lines to ensure that walls are vertical and floors are horizontal, which is essential for the structural integrity of a building.

3. Navigation:

   • Navigational systems rely on perpendicular lines to determine directions and plot courses. For example, latitude and longitude lines are perpendicular to each other.

Practice Problems

1. Find the equation of a line perpendicular to y = -3x + 4 and passing through the point (2, 5).
2. Find the equation of a line perpendicular to the line passing through points (-1, 3) and (2, -3) and passing through the point (0, -2).
3. Find the equation of a line perpendicular to 5x - 2y = 10 and passing through the point (-3, 1).
4. Find the equation of a line perpendicular to x = 4 and passing through the point (5, 2).
5. Find the equation of a line perpendicular to y = -2 and passing through the point (3, -4).

Solutions to Practice Problems

1. Problem: Find the equation of a line perpendicular to y = -3x + 4 and passing through the point (2, 5).

   • Solution:
     • The slope of the given line is m = -3.
     • The slope of the perpendicular line is m_perp = -1 / (-3) = 1/3.
     • Using the point-slope form: y - 5 = (1/3)(x - 2).
     • Simplifying to slope-intercept form: y = (1/3)x - 2/3 + 5 = (1/3)x + 13/3.
     • The equation of the perpendicular line is y = (1/3)x + 13/3.

2. Problem: Find the equation of a line perpendicular to the line passing through points (-1, 3) and (2, -3) and passing through the point (0, -2).

   • Solution:
     • The slope of the given line is m = (-3 - 3) / (2 - (-1)) = -6 / 3 = -2.
     • The slope of the perpendicular line is m_perp = -1 / (-2) = 1/2.
     • Using the point-slope form: y - (-2) = (1/2)(x - 0).
     • Simplifying to slope-intercept form: y = (1/2)x - 2.
     • The equation of the perpendicular line is y = (1/2)x - 2.

3. Problem: Find the equation of a line perpendicular to 5x - 2y = 10 and passing through the point (-3, 1).

   • Solution:
     • Rewrite the equation in slope-intercept form: -2y = -5x + 10 → y = (5/2)x - 5.
     • The slope of the given line is m = 5/2.
     • The slope of the perpendicular line is m_perp = -1 / (5/2) = -2/5.
     • Using the point-slope form: y - 1 = (-2/5)(x - (-3))
     • Simplifying to slope-intercept form: y = (-2/5)x - 6/5 + 1 = (-2/5)x - 1/5.
     • The equation of the perpendicular line is y = (-2/5)x - 1/5.

4. Problem: Find the equation of a line perpendicular to x = 4 and passing through the point (5, 2).

   • Solution:
     • The line x = 4 is a vertical line, so its slope is undefined.
     • A line perpendicular to a vertical line is a horizontal line.
     • The equation of a horizontal line passing through (5, 2) is y = 2.
     • The equation of the perpendicular line is y = 2.

5. Problem: Find the equation of a line perpendicular to y = -2 and passing through the point (3, -4).

   • Solution:
     • The line y = -2 is a horizontal line, so its slope is 0.
     • A line perpendicular to a horizontal line is a vertical line.
     • The equation of a vertical line passing through (3, -4) is x = 3.
     • The equation of the perpendicular line is x = 3.

Conclusion

Finding the equation of a perpendicular line is a crucial skill in coordinate geometry with numerous applications in various fields. By understanding the relationship between the slopes of perpendicular lines and following the steps outlined in this guide, you can confidently solve any problem involving perpendicular lines. Remember to practice regularly and double-check your work to avoid common mistakes. With a solid grasp of this concept, you'll be well-equipped to tackle more advanced topics in mathematics and related disciplines.