Perpendicular Slope
2 × −1/2 = −1 flip it, then change the sign
The perpendicular slope of a line is the negative reciprocal of its slope: turn the fraction upside down, then change the sign. A slope of 2 gives −1/2. A slope of −3/4 gives 4/3. For any slope that is defined and not zero, multiplying it by its perpendicular gives −1.
Quick reference: common slopes and their perpendiculars
If you came here for one number, it is probably in this table.
| Slope | Perpendicular slope | Slope | Perpendicular slope |
|---|---|---|---|
| 1 | −1 | −1 | 1 |
| 2 | −1/2 | −2 | 1/2 |
| 3 | −1/3 | −3 | 1/3 |
| 4 | −1/4 | −4 | 1/4 |
| 5 | −1/5 | −5 | 1/5 |
| 1/2 | −2 | −1/2 | 2 |
| 1/3 | −3 | −1/3 | 3 |
| 2/3 | −3/2 | −2/3 | 3/2 |
| 3/4 | −4/3 | −3/4 | 4/3 |
| 0 (horizontal) | undefined (vertical) | undefined (vertical) | 0 (horizontal) |
The bottom row is the odd one out and it is worth reading twice. Everything above it follows the flip-and-negate rule. That last pair does not, and the section further down explains why.
The rule, and why it is the negative reciprocal
Two lines are perpendicular when they cross at a right angle. Written as a rule, if their slopes are m1 and m2, then:
m1 × m2 = −1
The formula is often memorised without the reason behind it. It is worth a moment on why, because the reason makes it much harder to forget.
Slope is rise over run. Take a line with slope 3/4: from any point, go 4 across and 3 up. Now rotate that whole movement by a quarter turn, which is what making it perpendicular does. The 4 across becomes 4 up, and the 3 up becomes 3 to the left. So the new movement is 3 across and 4 up, but in the opposite horizontal direction, which is a slope of −4/3.
That is the whole thing. Rotating swaps the rise and the run, which is the reciprocal, and it reverses one of the two directions, which is the sign change. The two halves of the phrase “negative reciprocal” are just those two effects, and multiplying a number by its negative reciprocal always lands on −1 because the fractions cancel.
How to find a perpendicular slope, step by step
- Get the slope of the line you were given. If the equation is solved for y, it is the number multiplying x. If it is not, solve for y first.
- Write it as a fraction. A whole number like 5 is 5/1. This step is the one people skip, and it is what makes the next step obvious.
- Turn the fraction over. 5/1 becomes 1/5.
- Change the sign. 1/5 becomes −1/5.
Worked examples
- Slope 6. As a fraction, 6/1. Flipped, 1/6. Negated, −1/6.
- Slope −2/5. Flipped, −5/2. Negated, 5/2. A negative slope always has a positive perpendicular.
- From an equation. For y = 4x − 9, the slope is 4, so the perpendicular slope is −1/4. The −9 plays no part: the intercept moves a line up and down without tilting it.
- From two points. Through (1, 2) and (5, 10), the slope is (10 − 2) ÷ (5 − 1) = 2. The perpendicular slope is −1/2.
- When the equation is not solved for y. Take 3x + 2y = 6. Move the x term: 2y = −3x + 6. Divide every term by 2: y = −3/2 x + 3. Now the slope is visible as −3/2, so the perpendicular slope is 2/3. Doing this in your head from the original equation is where wrong answers come from: the 3 in 3x is not the slope.
Checking is quicker than the working: multiply the two slopes and you should get exactly −1. If you get 1, you flipped without changing the sign. If you get something that is not −1 at all, you have changed the sign without flipping. The perpendicular slope calculator does both steps and shows them.
The special case where the formula does not apply
A horizontal line and a vertical line are perpendicular. Anybody can see that. But their slopes do not multiply to −1, and no amount of flipping will make them.
A horizontal line has a slope of 0. Flip 0/1 and you get 1/0, which is a division by zero and is not a number. A vertical line has no slope to start with, so there is nothing to flip. The rule needs two real slopes, and this pair only has one.
One mistake in particular sends people looking for help. Asked for a line perpendicular to y = 3, the reflex is to treat the 3 as a slope, flip it and answer −1/3. It is not a slope. In y = 3 there is no x term at all, so the slope is 0 and the line is flat.
The perpendicular is a vertical line, written x = a number, and which number depends on the point it has to pass through. Perpendicular to y = 3 through the point (20, 19) is x = 20. Not −1/3, and not y = anything.
The check that catches it: before flipping anything, look at whether the equation actually contains an x. If it does not, you are dealing with a horizontal line and the answer is vertical.
When perpendicular lines do not look perpendicular
This one causes more confusion than the arithmetic ever does, and it is easy to overlook. Two lines whose slopes multiply to −1 can look nothing like a right angle on a graph.
The reason is the grid, not the maths. Most graphing tools stretch the axes independently to fit whatever numbers you gave them, so one unit across is not the same number of pixels as one unit up. The moment the two scales differ, every drawn angle is distorted, and a perfectly good perpendicular pair can meet at what looks like 50 degrees or 120.
So a picture cannot settle the question. The arithmetic can. If the slopes multiply to −1, the lines are perpendicular, whatever the screen shows. If they do not, the lines are not perpendicular, however convincing the picture looks.
We hit this building our own tools. The right angle marker on our perpendicular calculator would not draw square, and the numbers had been right the whole time: the grid was fitting each axis separately, so the marker came out as a lopsided diamond. Forcing both axes onto one scale fixed it, which is why our graph page carries an equal axis scales switch you can turn off and watch a right angle stop being one.
If a graph ever makes you doubt a perpendicular pair, check whether both axes share a scale before you doubt the arithmetic.
Writing the whole equation, not just the slope
Many questions do not stop at the slope. They ask for the equation of the perpendicular line through a particular point, and that needs one more step.
The slope on its own describes a direction, not a line. There are infinitely many lines with a slope of −1/2, stacked above and below each other. The point is what picks one of them.
Worked example
Find the line perpendicular to y = 2x + 5 that passes through (4, 3).
- Slope of the given line: 2.
- Perpendicular slope: flip to 1/2, negate to −1/2.
- Use the point. Substitute x = 4, y = 3 and m = −1/2 into y = mx + b: 3 = −1/2(4) + b, so 3 = −2 + b, so b = 5.
- Write it: y = −1/2 x + 5.
Check by substituting the point back in: −1/2 × 4 + 5 = 3. It fits, and the two slopes multiply to −1, so both halves of the question are satisfied.
That the intercept came out as 5 in both equations here is a coincidence of the numbers, not a rule. Change the point to (4, 9) and the answer becomes y = −1/2 x + 11 while the given line is untouched.
Parallel slopes, and telling the two apart
Parallel is the easier of the pair. Parallel lines have exactly the same slope. No flipping, no sign change. A line parallel to y = 4x − 9 has a slope of 4, and the only thing that differs is where it crosses the axis.
| Parallel | Perpendicular | |
|---|---|---|
| The slopes | Identical | Negative reciprocals |
| Multiply them | m2, whatever that is | −1, when both slopes are defined |
| Angle where they meet | They never meet | 90 degrees |
| Example, from slope 2 | 2 | −1/2 |
| Vertical lines | Parallel to each other | Perpendicular to horizontal ones |
To tell which relationship two given lines have, read both slopes and compare. Same slope with different intercepts, parallel. Same slope and the same intercept, and it is one line written twice. Product of −1, perpendicular. Anything else, and they simply cross at some other angle. If you are working from two points rather than equations, find both slopes first with a rise over run calculator and then compare them.
One thing worth saying because it catches people out: two lines with the same slope and the same intercept are not parallel in any useful sense. They are the same line drawn twice.
Practice questions, with the answers
- What is the perpendicular slope of 7?
- What is the perpendicular slope of −3/8?
- A line has the equation y = −5x + 2. What is the slope of a line perpendicular to it?
- Are the lines y = 2x + 1 and y = 2x − 6 parallel, perpendicular or neither?
- What is the slope of a line perpendicular to y = −4?
- Write the equation of the line perpendicular to y = 3x − 1 that passes through (6, 4).
Show the answers
- −1/7. Write 7 as 7/1 first if it helps.
- 8/3. Flip to −8/3, then change the sign.
- 1/5. The slope is −5, so flip to −1/5 and negate.
- Parallel. Both slopes are 2, and 2 × 2 is 4, not −1.
- Trick question, and the important one. y = −4 is horizontal with slope 0. The perpendicular line is vertical, so its slope is undefined rather than being some number you can calculate. Its equation is x = a number, and which number depends on the point it has to pass through.
- Perpendicular slope is −1/3. Then 4 = −1/3(6) + b gives 4 = −2 + b, so b = 6. The equation is y = −1/3 x + 6.
Frequently asked questions
How do I know if two lines are parallel or perpendicular?
Compare their slopes. The same slope means parallel. Slopes that multiply to −1 mean perpendicular. Anything else means they cross at some angle that is neither, which is a perfectly valid answer to a question that asks.
If the equations are not solved for y, do that first. Comparing the coefficients of two equations written in different shapes is where wrong answers come from.
Does the y-intercept matter when finding a perpendicular slope?
Not at all. The intercept slides a line up and down without changing how steep it is, and perpendicularity is entirely about steepness. Every line with a slope of 3 has the same perpendicular slope of −1/3, wherever it sits on the grid.
The intercept only comes back in when you are asked for the full equation of the perpendicular line rather than just its slope, because then you need a point it passes through.
Why is it negative and not just the reciprocal?
Because a quarter turn reverses one direction as well as swapping the two. Flipping alone would give you a line that is steeper or shallower in the same general direction, which is not a right angle.
There is a quick sanity check built into it: one of any perpendicular pair climbs and the other falls. If both of your slopes are positive, or both negative, they cannot be perpendicular whatever the arithmetic says.
Can two perpendicular lines both have whole number slopes?
Only one pair: 1 and −1. For any other whole number the reciprocal is a fraction, so the perpendicular slope is a fraction too. That is why perpendicular answers so often look untidy next to the slope they came from, and why an answer that stays whole is worth double checking.
Do parallel lines always have the same slope?
Yes, with one wrinkle. Two vertical lines are parallel to each other, but neither has a slope to compare, so the test has to be done by looking rather than by arithmetic. Every other pair of parallel lines has two equal, defined slopes.