Perpendicular Lines

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Perpendicular Linesvideo

Perpendicular Lines

Perpendicular lines

What are perpendicular lines?

Perpendicular lines are straight lines which meet at right-angles (90°)
One line may be referred to as a normal to the other line

How are the gradients of perpendicular lines related?

Gradients m₁ and m₂ are perpendicular if m₁ × m₂ = −1
- For example
  - 1 and −1
  - $\frac{1}{3}$ and -3
  - $- \frac{2}{3}$ and $\frac{3}{2}$
The two gradients are negative reciprocals of one another
We can use $m_{2} = - \frac{1}{m_{1}}$ to find a perpendicular gradient

How can I tell if two lines are perpendicular?

Given two lines in the form $y = m x + c$ , simply check if their gradients $(m)$ are negative reciprocals of one another
- $y = \frac{1}{3} x + 10$ and $y = - 3 x - 18$ are perpendicular
- $y = \frac{1}{7} x + 16$ and $y = 7 x - 8$ are not perpendicular
One or both of the equations may not be written in the form $y = m x + c$
- In this case, you should rearrange both equations into the form $y = m x + c$
- Their gradients can then be easily compared

How do I find the equation of a line perpendicular to another?

You need to be able to find the equation of line that passes through a particular point and is perpendicular to another line
- E.g. $5 y = 4 x + 30$ which passes through the point (8, 3)
Rearrange the equation into the form $y = m x + c$ so that its gradient can be identified more easily
- $y = \frac{4}{5} x + 6$
Find the gradient of the perpendicular line
- The gradient of the original line is $\frac{4}{5}$
- Therefore the gradient of the perpendicular line is $- \frac{5}{4}$
- The perpendicular line has an equation in the form $y = - \frac{5}{4} x + c$
Substitute the given point into the equation for the perpendicular and solve for $c$
- Substitute (8, 3), into $y = - \frac{5}{4} x + c$
- $3 = - \frac{5}{4} (8) + c$
- $c = 13$
Substitute the value of $c$ to find the equation of the perpendicular
- The equation of the perpendicular line is $y = - \frac{5}{4} x + 13$
  - This could also be written as $4 y = - 5 x + 52$ or equivalent

Worked example

The line L has equation $y - 2 x + 2 = 0$ .

Find the equation of the line perpendicular to L which passes through the point $(2, - 3)$ .

Leave your answer in the form $a x + b y + c = 0$ where $a$ , $b$ and $c$ are integers.

Answer:

Rearrange L into the form $y = m x + c$ so we can identify the gradient

$\begin{array}{rcl} y - 2 x + 2 & = & 0 \\ y & = & 2 x - 2 \end{array}$

Gradient of L is 2

The gradient of the line perpendicular to L will be the negative reciprocal of 2

$m = - \frac{1}{2}$

Substitute the point $(2, - 3)$ into the equation $y = - \frac{1}{2} x + c$
Solve for $c$

$\begin{array}{rcl} y & = & - \frac{1}{2} x + c \\ - 3 & = & - \frac{1}{2} (2) + c \\ - 3 & = & - 1 + c \\ c & = & - 2 \end{array}$

Write the full equation of the line

$\begin{array}{rcl} y & = & - \frac{1}{2} x - 2 \end{array}$

The question asks for the line to be written in the form $a x + b y + c = 0$ where $a$ , $b$ and $c$ are integers

Move all the terms to the left hand side

$\frac{1}{2} x + y + 2 = 0$

Then multiply every term by 2, to ensure they are all integers

$x + 2 y + 4 = 0$

How do I find the equation of a perpendicular bisector?

A perpendicular bisector of a line segment cuts the line segment in half at a right angle
Finding the equation of the perpendicular bisector of a line segment is very similar to finding the equation of a any perpendicular
- Find the coordinates of the midpoint of the line segment
  - The perpendicular bisector will pass through this point
- Find the gradient of the line segment
- Then find the negative reciprocal of this gradient
  - This will be the gradient of the perpendicular bisector, $m$
- Write the equation of the perpendicular bisector in the form $y = m x + c$
- Substitute the midpoint of the line segment into the equation of the perpendicular bisector
  - Solve to find $c$
- Write the full equation of the perpendicular bisector in the form $y = m x + c$
- Rearrange the equation if the question requires a different form

Worked example

Find the equation of the perpendicular bisector of the line segment joining the points (4, -6) and (8, 6).

Answer:

Find the coordinates of the midpoint of the line segment
The perpendicular bisector will pass through this point

$(\frac{4 + 8}{2}, \frac{- 6 + 6}{2}) = (6, 0)$

Find the gradient of the line segment

$\frac{- 6 - 6}{4 - 8} = \frac{- 12}{- 4} = 3$

Find the negative reciprocal of this
This will be the gradient of the perpendicular bisector, $m$

$m = - \frac{1}{3}$

Write the equation of the perpendicular bisector in the form $y = m x + c$

$y = - \frac{1}{3} x + c$

Substitute in the midpoint (6, 0) and solve to find $c$

$\begin{array}{rcl} 0 & = & - \frac{1}{3} (6) + c \\ 0 & = & - 2 + c \\ c & = & 2 \end{array}$

Write the full equation of the perpendicular bisector

$y = - \frac{1}{3} x + 2$