Finding Perpendicular Equations- Algebra Guide
What Perpendicular Lines Actually Are
Two lines are perpendicular when they intersect at a right angle (90 degrees). That's it. No fancy geometry jargon needed.
In coordinate geometry, perpendicular lines have a specific mathematical relationship. Their slopes multiply to give -1. This is the foundation of everything that follows.
The Negative Reciprocal Rule
If a line has slope m, a perpendicular line has slope -1/m.
Let's break this down:
- Take your original slope
- Flip it (reciprocal)
- Change the sign (negative)
Example: If your slope is 2, the perpendicular slope is -1/2.
Example: If your slope is -3/4, the perpendicular slope is 4/3.
Example: If your slope is 5, the perpendicular slope is -1/5.
How to Find the Equation of a Perpendicular Line
Step 1: Find the Slope of Your Original Line
Convert your equation to slope-intercept form: y = mx + b
The coefficient of x is your slope (m).
Step 2: Calculate the Perpendicular Slope
Flip and negate. If original slope is 3, perpendicular slope is -1/3.
Step 3: Plug Into Point-Slope Form
Use the point your perpendicular line must pass through:
y - y₁ = m(x - x₁)
Substitute your perpendicular slope for m, and your point's coordinates for (x₁, y₁).
Step 4: Simplify
Distribute and rearrange into whatever form your assignment requires—slope-intercept, standard, or point-slope.
Worked Example
Problem: Find the equation of the line perpendicular to y = 2x + 5 that passes through (3, 1).
Step 1: Original slope is 2.
Step 2: Perpendicular slope = -1/2.
Step 3: y - 1 = -1/2(x - 3)
Step 4: y - 1 = -1/2x + 3/2
y = -1/2x + 5/2 ✓
Finding Perpendicular Equations Through Two Points
Sometimes you don't have a pre-made line. You need to find a line perpendicular to one passing through two given points.
Problem: Find the line perpendicular to the segment connecting (2, 4) and (6, 8) that passes through (4, 2).
Step 1: Find slope of segment: (8-4)/(6-2) = 4/4 = 1
Step 2: Perpendicular slope = -1
Step 3: y - 2 = -1(x - 4)
Step 4: y = -x + 6 ✓
Common Mistakes That Will Cost You Points
- Forgetting to flip AND negate. Just negating gives you a parallel line, not perpendicular.
- Mixing up the original slope. Make sure you're working with the correct line's slope.
- Arithmetic errors. Fractions are where most people mess up. Double-check your calculations.
- Using the wrong point. The perpendicular line must pass through the specified point, not just any point.
Perpendicular vs. Parallel: Quick Comparison
| Relationship | Slope Connection | Example (if m = 3) |
|---|---|---|
| Perpendicular | m × m₂ = -1 | m₂ = -1/3 |
| Parallel | m = m₂ | m₂ = 3 |
Quick Reference
- Horizontal lines (slope 0) → Perpendicular lines are vertical (undefined slope)
- Vertical lines (undefined slope) → Perpendicular lines are horizontal (slope 0)
This exception trips people up. Vertical and horizontal lines are perpendicular to each other, but you can't use the negative reciprocal formula on them.
Practice Problems to Try
- Find the equation perpendicular to y = -4x + 7 through (2, 3)
- Find the line perpendicular to 2x + 3y = 6 through (-1, 4)
- Find the perpendicular bisector of the segment from (1, 2) to (5, 6)
Check your answers by verifying that the slopes multiply to -1.