Math
Perpendicular Line Calculator
Enter any line and a point, and mathcheck writes the perpendicular line through that point with exact fractions in slope-intercept, point-slope, standard, and general form.
It shows the negative-reciprocal step, the right-angle intersection point, the distance from the point to the given line, and a graph of both lines.
Type a fraction like 3/2 or a decimal like -0.5. Type "undefined" for a vertical line.
Where the line crosses the y-axis.
x-coordinate of the point on the perpendicular line.
y-coordinate of the point on the perpendicular line.
Try an example
Result
Perpendicular line
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- Given line
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- Perpendicular slope
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- Given slope
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- Intersection point
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- Standard form
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Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What makes two lines perpendicular?
Two lines are perpendicular when they meet at a 90° angle. For non-vertical lines this happens exactly when their How steep a line is: how much y changes for each step right in x. are negative The number you get by flipping a fraction; 3/4 becomes 4/3, and 5 becomes 1/5.: m1 × m2 = -1. Flip the fraction and change the sign. A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope).
Worked example
Perpendicular to y = 2x + 1 through (3, 4)
The given slope is 2, so the perpendicular slope is -1/2. Point-slope: y - 4 = (-1/2)(x - 3). Distribute: y - 4 = (-1/2)x + 3/2, so y = (-1/2)x + 11/2. The lines cross at (9/5, 23/5).
| Given slope | Perpendicular slope | Why |
|---|---|---|
| 2 | -1/2 | flip 2/1 to 1/2, change the sign |
| -3/4 | 4/3 | flip 3/4, change the sign |
| 1 | -1 | flip 1/1, change the sign |
| 0 (horizontal) | undefined (vertical) | no reciprocal of 0; the perpendicular is vertical |
| undefined (vertical) | 0 (horizontal) | the perpendicular is horizontal |
How to use this calculator
- Choose how the given line is written and enter it.
- Enter the point the perpendicular line must pass through.
- Read the perpendicular line's equation in the header; the given line appears in the box below it.
- Check the grid for the two slopes (their product is -1) and the intersection point where the right angle sits.
- Open Show the work to see the negative reciprocal and the point-slope steps.
How to read the answer
The result is the unique line through your point that meets the given line at 90°. Its slope is the negative reciprocal of the given slope, and the intersection point is the foot of the perpendicular, the closest point on the given line to your point. The distance shown is exactly the length of that perpendicular segment.
Common mistakes and edge cases
- Only flipping the fraction or only changing the sign. The perpendicular slope needs both: 2/3 becomes -3/2.
- Using the same slope, which gives a parallel line instead.
- Trying to take the reciprocal of 0: the perpendicular to a horizontal line is vertical, x = x0, with no slope-A point where a graph crosses an axis. form.
- Reusing the original y-intercept. Solve for the new intercept with the given point.
- Forgetting that the perpendicular bisector of a segment needs the The point exactly halfway between two other points. as the point, not an endpoint.
Worked examples
Perpendicular to y = 2x + 1 through (3, 4)
Perpendicular line
y = (-1/2)x + 11/2
Negative fraction slope
Perpendicular to y = (-3/4)x + 2 through (0, -1)
Perpendicular line
y = (4/3)x - 1
Given line in standard form
Perpendicular to 3x - 2y = 6 through (1, 1)
Perpendicular line
y = (-2/3)x + 5/3
Perpendicular bisector setup
Perpendicular to the line through (1, 1) and (5, 3), through the midpoint (3, 2)
Perpendicular line
y = -2x + 8
Horizontal given line
Perpendicular to y = 5 through (2, -1)
Perpendicular line
x = 2
Vertical given line
Perpendicular to x = 4 through (-2, 3)
Perpendicular line
y = 3
Identical points for the given line
Perpendicular line
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Frequently asked questions
What is the slope of a line perpendicular to y = mx + b?+
The negative reciprocal, -1/m. For example, the perpendicular slope to 2/3 is -3/2, and the product (2/3)(-3/2) = -1.
How do I find a perpendicular line through a point?+
Find the negative reciprocal of the given slope, then use point-slope form with your point: y - y0 = (-1/m)(x - x0). Solve for y to get slope-intercept form.
What is perpendicular to a horizontal line?+
A vertical line. The slope 0 has no reciprocal, so the perpendicular through (x0, y0) is simply x = x0.
How do I find the perpendicular bisector of a segment?+
Find the midpoint of the segment and the slope of the segment, then build the perpendicular line through the midpoint. Enter the segment's endpoints as the two points and the midpoint as the point the new line passes through.
Where do the two lines intersect?+
At the foot of the perpendicular, the closest point on the given line to your point. The calculator shows it as an exact fraction pair, and the distance from your point to that foot is the point-to-line distance.
Can two lines with positive slopes be perpendicular?+
No. Perpendicular slopes multiply to -1, so one must be positive and the other negative (or one is 0 and the other undefined).
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026