Equation of a perpendicular line
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
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.