Math calculator
Both lines through your point — including the awkward two.
Including the two cases the rule does not cover.
original slope 2
y = −1/2x + 1.5
The perpendicular slope is the negative reciprocal, and the two multiply to −1.
Both lines through your point
| Line | Slope | Slope-intercept | Standard |
|---|---|---|---|
| Parallel | 2 | y = 2x − 1 | 2x − y = 1 |
| Perpendicular | −1/2 | y = −1/2x + 1.5 | x + 2y = 3 |
Parallel slope
2
the same as the original
Perpendicular slope
−1/2
the negative reciprocal
Product of slopes
−1
always −1 when both exist
Both pass through
(1, 1)
your point
Horizontal and vertical are handled explicitly, because 0 × undefined is not −1 and the product rule genuinely fails there.
Parallel keeps the slope; perpendicular flips it and changes the sign. The rule that they multiply to −1 covers almost everything — and fails exactly where a horizontal line meets a vertical one, because 0 × undefined is not −1.
The horizontal and vertical cases are handled, not assumed away.
Updated 7 September 2026 · Works in any browser, no installation
Parallel keeps the slope; perpendicular takes the negative reciprocal. A line of slope 2 has parallels of slope 2 and perpendiculars of slope −1/2. Flip the fraction, change the sign, and substitute your point — both steps are needed, and dropping the sign is the usual error.
At a glance
Two lines never meet exactly when they climb at the same rate. That is the whole condition: m₁ = m₂.
It is why slope is the useful quantity here rather than the angle. Equal angles would work too, but slopes compare as numbers and angles need a trigonometric function first.
Two vertical lines are also parallel, and neither has a slope. The slope test cannot see that case at all — it has to be handled separately, which is a recurring theme on this page.
Two steps, and both are needed. Flip the fraction, then change the sign.
The perpendicular to 2/3 is −3/2. Flipping alone gives 3/2, which is a line at a different angle entirely; negating alone gives −2/3, which is its mirror image and also not perpendicular.
Why the product is −1: rotating a line by 90° turns a rise of a and a run of b into a rise of b and a run of −a. The new slope is −b/a, and multiplying by a/b gives −1. The rule is a rotation written in coordinates.
A useful consequence: slope 1 and slope −1 are perpendicular, which is why the diagonals of a square are at right angles.
A horizontal line has slope 0. A vertical line has no slope. They are perpendicular to each other — obviously, they are the axes — and the product rule cannot say so.
0 × undefined is not −1. It is not anything. The rule assumes both slopes exist, and in this one pair they do not.
Applied blindly it produces nonsense. −1/0 in floating point is −Infinity, which then flows into y = mx + c and gives NaN a few steps later. This page branches on the case instead and reports the perpendicular to a horizontal line as vertical, with its x = k equation.
Worth remembering as a pair: the exceptions to the perpendicularity rule are exactly the two lines that are perpendicular most obviously.
Same slope and a different intercept: parallel. Same slope and the SAME intercept: the same line.
Whether a line counts as parallel to itself is a convention rather than a fact, and textbooks split on it. Most school geometry says no — parallel lines are distinct and never meet — while some higher treatments say yes, because it makes parallelism an equivalence relation.
This page always produces a line through your point, which is the same line whenever your point already lies on the original. That is the honest output, and it is worth noticing rather than hiding.
In a system of equations the distinction is the whole answer: same slope and different intercept means no solution, same slope and same intercept means infinitely many.
Given two equations rather than a slope and a point, get both slopes and compare.
From y = mx + c: the slope is the m, directly.
From Ax + By = C: the slope is −A/B. The minus sign is the step people drop, and dropping it turns a parallel test into a perpendicular one.
Then: equal slopes means parallel, a product of −1 means perpendicular, and neither means they cross at some other angle. B = 0 on either line means that line is vertical, and the special cases apply.
Shortest distance to a line. Drop a perpendicular from the point to the line and measure it. Constructing that perpendicular is exactly this calculation.
Proving a shape. A rectangle needs adjacent sides perpendicular; a parallelogram needs opposite sides parallel. Both reduce to comparing slopes.
Construction and drawing. Setting out a wall parallel to an existing one, or a partition at right angles to it.
Graphics. A surface normal is perpendicular to a surface, and in two dimensions finding it is the negative reciprocal — which is why reflections and lighting calculations keep meeting this rule.
The slope conditions are curriculum standards; these are the references.
Method. The perpendicular slope is the exact negative reciprocal of the original, computed on rationals, and the two degenerate cases are branched on rather than derived: a vertical original gives a horizontal perpendicular of slope 0, and a horizontal original gives a vertical perpendicular with no slope at all. The product of the two slopes is reported only when both exist, so the page never prints a product for a pair the rule does not cover. The suite substitutes the given point into both new lines in standard form on integers. That engine is verified on every change against 67 hand-written assertions, including that the parallel keeps the original slope exactly and that both new lines pass through the given point, checked in whole numbers across three thousand generated cases. The count and the per-case breakdown are published on the formula verification page.
To write either line in every standard form, or to start from two points rather than a slope, the Line Equation Calculator covers that.
Educational use disclaimer
This is an educational tool. Slopes are exact for what you enter; the horizontal and vertical cases are handled explicitly rather than through a rule that does not cover them.
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Published 7 September 2026