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# Equation of a perpendicular line
- URL: https://www.esheets.io/equation-of-a-perpendicular-line/
- Published: 2024-09-24T20:06:13.000Z
- Updated: 2026-06-21T16:42:03.000Z
- Author: Richard Linnington
- Tags: Maths, Algebra

In geometry, finding the equation of a perpendicular line is useful when working with shapes like squares and rectangles or analyzing graphs. Perpendicular lines meet at a right angle (90 degrees), and their slopes are opposite reciprocals. This concept helps in tasks such as determining how to build structures with perfect corners or understanding angles in real-world designs like roads and bridges. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Practise the equation of a perpendicular line with this self-marking maths worksheet.

The interactive worksheet below generates questions, gives instant feedback, and lets students record their score.

#### What you’ll practise

- Recognising that perpendicular gradients are negative reciprocals.
- Finding the perpendicular gradient.
- Using the given point on the new line.
- Writing the equation of the perpendicular line.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Enter the equation of a perpendicular line in the form y = mx + c. For the gradient, enter the correct value or fraction, and for the intercept, input any number.

  
## Topic guide

### What this worksheet practises

This worksheet provides practice on finding the equation of a line that is perpendicular to another given line. Perpendicular lines intersect at exactly 90 degrees (a right angle). This is a higher-level geometry skill that relies heavily on understanding negative reciprocals.

### Key method

If two lines are perpendicular, their gradients multiply together to make −1\. In simpler terms, the gradient of the perpendicular line is the **negative reciprocal** of the original gradient.

- Find the gradient of the original line. Let's call it 'm'.
- Find the negative reciprocal. Flip the number upside down (as a fraction) and change its sign. This gives you the new gradient.
- Start writing your new equation: y = (new gradient)x + c.
- Substitute the given coordinate point into the equation to calculate 'c'.
- Write out the final complete equation.

### Worked example

**Find the equation of the line perpendicular to y = 2x + 5 that passes through the point (6, 1).**

Step 1: Find the new gradient. The original gradient is 2 (which is 2/1).

Flip it to 1/2, and change the sign to negative. The new gradient is −1/2.

Our equation is y = −1/2 x + c.

Step 2: Substitute the coordinate (6, 1) to find 'c'.

1 = −1/2(6) + c

1 = −3 + c

c = 1 + 3 = 4.

Step 3: Write the final equation.

y = −1/2 x + 4.

### Common mistakes to avoid

The most common error is only doing half of the negative reciprocal rule: either flipping the fraction but forgetting to change the sign, or changing the sign but forgetting to flip the fraction. Remember it takes two steps: flip the number, and flip the sign.

### How to check your answer

To check you have the correct perpendicular gradient, multiply your two gradients together. In our example, 2 × (−1/2) = −1\. Because the result is −1, you can be 100% certain the two lines cross at right angles.