In short: To find the equation of a perpendicular line, take the negative reciprocal of the original gradient (flip the fraction and change its sign). Then substitute the given point into y = mx + c with that new gradient to find c, and write the final equation.
Finding the equation of a perpendicular line is a higher-tier GCSE maths skill (AQA) that hinges on one rule: perpendicular gradients multiply to −1. This guide shows how to get the negative reciprocal, how to finish with a substitution, and a worked example with every step checked, plus the errors that cost marks.
The reliable method
You are typically given a line and a point, and asked for the perpendicular line through that point. Follow these steps.
- Find the gradient of the original line. Rearrange into y = mx + c if needed and read off m.
- Take the negative reciprocal. The perpendicular gradient is −1 ÷ m. In practice, flip the fraction and change the sign. For a gradient of 2 (that is 2/1), the perpendicular gradient is −1/2.
- Substitute the given point. Put the point's x and y into y = mx + c using the new perpendicular gradient. Only c is unknown.
- Solve for c and write the equation. Find the intercept, then state the full equation y = mx + c.
A quick check: the two gradients should multiply to give −1. If they do, the lines are perpendicular.
A worked example
Find the equation of the line perpendicular to y = 2x + 3 that passes through the point (4, 1).
Step 1 — gradient of the original. The line y = 2x + 3 has gradient m = 2.
Step 2 — negative reciprocal. Flip 2 (which is 2/1) to get 1/2, then change the sign:
perpendicular gradient = −1/2
Step 3 — substitute (4, 1). Use x = 4 and y = 1 with gradient −1/2:
1 = −1/2 × 4 + c 1 = −2 + c
Step 4 — solve for c.
c = 1 + 2 = 3
So the perpendicular line is y = −1/2 x + 3.
Check. The gradients multiply: 2 × (−1/2) = −1, confirming they are perpendicular. Substituting (4, 1) gives −1/2 × 4 + 3 = −2 + 3 = 1, which matches.
This works because perpendicular lines meet at a right angle exactly when their gradients multiply to −1. The negative reciprocal guarantees that product, and the given point fixes the one perpendicular line passing through it.
Common mistakes to avoid
- Forgetting the negative sign. The reciprocal alone is not enough. The gradient of 2 gives a perpendicular gradient of −1/2, not 1/2. Change the sign every time.
- Only changing the sign. Equally, just negating to −2 is wrong — you must also flip the fraction to take the reciprocal.
- Mishandling fractional gradients. The negative reciprocal of 3/4 is −4/3: flip and change sign. Practise this with fractions, not just whole numbers.
- Reusing the original gradient. Perpendicular is not parallel. Do not keep the same gradient; transform it with the negative reciprocal.
- Sign slip solving for c. Solving 1 = −2 + c gives c = 1 + 2 = 3. Moving −2 across the equals sign makes it +2.
Frequently asked questions
How do you find the gradient of a perpendicular line? Take the negative reciprocal of the original gradient: flip the fraction and change its sign. If the original gradient is m, the perpendicular gradient is −1 ÷ m. For example, a gradient of 4 gives a perpendicular gradient of −1/4.
Why do perpendicular gradients multiply to −1? When two lines cross at a right angle, rotating one by 90 degrees turns its rise into a run and its run into a rise, with a sign change. Algebraically this means the product of their gradients is always −1.
What is the perpendicular gradient of a horizontal line? A horizontal line has gradient 0, and its perpendicular is a vertical line. Vertical lines have an undefined gradient and are written as x = a number, because the negative reciprocal of 0 does not exist.
Do I keep the same y-intercept as the original line? No. You recalculate c by substituting the given point into y = mx + c with the new perpendicular gradient. The intercept will usually differ from the original.
How do I check my perpendicular line is right? Multiply the two gradients — they should give −1 — and substitute the given point into your equation to confirm both sides are equal.
Practise this
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