The perpendicular rule is one of those results that is easy to memorise and easy to apply wrongly. Flip the fraction, change the sign — students remember there are two steps but frequently perform only one. Understanding why the rule holds makes it much harder to get wrong.
The Rule
Two lines meet at a right angle exactly when their slopes are negative reciprocals. The word carries both operations: reciprocal means flip the fraction, negative means change the sign.
| Original slope | Perpendicular slope |
|---|---|
| undefined (vertical) |
Whole numbers catch people out. A slope of
Why the Product Is −1
This is not an arbitrary fact. Turn a line through 90° and watch what happens to its slope triangle.
Take a line of slope
The rise and the run have swapped roles, and exactly one of them has changed sign. That is precisely what “negative reciprocal” describes, and multiplying confirms it:
Because exactly one sign flips, a perpendicular pair always has one positive and one negative slope — unless one of them is horizontal or vertical. If you compute a perpendicular slope and it has the same sign as the original, you have made an error.
Writing a Perpendicular Line Through a Point
Two steps: flip the slope, then use point-slope form with the given point.
Write the line perpendicular to
The original slope is
If the original is in standard form, there is a neat shortcut: swap
The Exception: Horizontal and Vertical
A horizontal line and a vertical line clearly meet at a right angle. But the slope rule breaks down:
- Horizontal: slope
. - Vertical: slope undefined.
You cannot multiply
So the complete statement is: two lines are perpendicular if their slopes multiply to
Perpendicular Bisectors
A perpendicular bisector cuts a segment in half at a right angle. It needs two ingredients, and the name tells you both.
- Bisector — it passes through the midpoint:
. - Perpendicular — its slope is the negative reciprocal of the segment’s slope.
For the segment from
- Midpoint:
. - Segment slope:
, so the bisector’s slope is . - Equation:
.
Every point on a perpendicular bisector is the same distance from both endpoints — which is why this construction turns up when finding the centre of a circle through three points.
Worked Example A: Flip the Slope
Find the slope perpendicular to each:
Each time the sign changed and the fraction inverted.
Worked Example B: Through a Point
Write the line perpendicular to
Perpendicular slope:
Worked Example C: From Standard Form
Write the line perpendicular to
Swap and change one sign: the perpendicular family is
Check by slopes: the original has slope
Worked Example D: Is the Triangle Right-Angled?
Triangle with vertices
: : :
Test the products:
Worked Example E: Perpendicular Bisector
Find the perpendicular bisector of the segment from
- Midpoint:
. - Segment slope:
. - Bisector slope:
.
Common Mistakes to Avoid
- Flipping without changing the sign. Perpendicular to
is , not . - Changing the sign without flipping. Nor is it
; that is a different line, and parallel to the original’s mirror image. - Forgetting a whole number is a fraction. Perpendicular to
is , not . - Applying the rule to a horizontal line. Slope
has no reciprocal; the perpendicular is vertical, written . - Reading the slope before solving for
. has slope , so its perpendicular slope is . - Using the original point on the wrong line. The given point belongs to the new line, not the one you were given.
- Assuming a right angle from a sketch. Slopes of
and look perpendicular but multiply to .
Where This Shows Up
- Perpendicular bisectors, used to find circumcentres and the centre of a circle through three points.
- Shortest distance from a point to a line, which is measured along the perpendicular.
- Geometry proofs involving rectangles, squares, rhombuses and right-angled triangles.
- Normal lines in calculus — the normal at a point is perpendicular to the tangent there.
- Computer graphics and physics, where a surface normal determines how light reflects or how an object bounces.
Practice Problems
Work each one before opening the answer.
Problem 1. What slope is perpendicular to
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Problem 2. What slope is perpendicular to
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Problem 3. Are
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Problem 4. Are
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Problem 5. Write the line perpendicular to
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Slope
Problem 6. Write the line perpendicular to
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Slope
Problem 7. Write the line perpendicular to
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Swap and change one sign:
Problem 8. What line is perpendicular to
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Problem 9. Find the perpendicular bisector of the segment from
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Midpoint
Problem 10. For what
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Problem 11. Is the triangle
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Problem 12. Lines
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Quick Reference
| Question | Answer |
|---|---|
| Perpendicular condition | |
| Finding the slope | Flip the fraction and change the sign |
| Whole number | Perpendicular slope is |
| Sign check | One slope positive, one negative |
| Through a point | Flip the slope, then use point-slope |
| Standard-form shortcut | Swap |
| Horizontal line | Perpendicular is vertical, |
| Perpendicular bisector | Midpoint + negative reciprocal slope |
The matching rule for lines that never meet is in Parallel Lines. Both depend on slope and appear together in Lines; perpendicular bisectors lead directly into Circles. More Algebra lessons are available.