How this instrument works
A line perpendicular to another one meets it at a right angle, and slope carries that entire geometric fact in a single number: m_perp = −1 ⁄ m. A parallel line, by contrast, keeps the original slope completely unchanged — same number, same direction. A perpendicular line instead flips that number upside down and reverses its sign, two separate operations, not one. Skip either step — invert only, or negate only — and the result crosses at some other angle, not ninety degrees.
The negative reciprocal falls out of a simple rotation. A line with slope m runs 1 unit right for every m units up, so its direction is the vector (1, m). Rotate that vector a quarter turn and it becomes (−m, 1) — run and rise swap places, and one of them changes sign. Read the rotated vector back as a slope, rise over run, and the result is 1 ⁄ (−m), the same as −1 ⁄ m. The two numbers will always multiply to −1, since a slope times its own negative reciprocal cancels exactly, true for any starting value that is not zero.
The one input this identity refuses is m = 0. A horizontal line has no rise at all, so its perpendicular partner is a vertical line — straight up and down, with a slope that is undefined rather than a reciprocal of zero. Vertical lines cannot be written as y = mx + b at all, since a single x-value would map to infinitely many y-values, which is why this sheet flags that case rather than returning a number that would only mislead.
- Enter the original line's steepness into Original slope, m — this is the slope you already know.
- Enter the shared point's coordinates into Point: x and Point: y — the point the new, perpendicular line must pass through.
- Read Perpendicular slope — it is always the negative reciprocal of the original, computed automatically from your entries.
- Read Perpendicular line's y-intercept, b, and pair it with the perpendicular slope in y = mx + b to write the full equation.
Worked example — slope 2 through (4, 3)
A ramp's main rail runs with slope 2, and a diagonal support brace needs to cross it at a perfect right angle while passing through the point (4, 3). The perpendicular slope is m_perp = −1 ⁄ 2 = −0.5 — flip the slope upside down, then flip its sign. Checking the product confirms it: 2 × (−0.5) = −1, the algebraic signature of a true right angle. To locate the brace's own y-intercept, plug the shared point back into b = y₁ − m_perp·x₁ = 3 − (−0.5 × 4) = 3 + 2 = 5.
The brace's full equation is therefore y = −0.5x + 5, a line that drops half a unit for every unit it runs right — the opposite lean of the original rail, which climbs two units for every one it runs right. Multiply the two slopes, 2 and −0.5, and the −1 product is not a coincidence: it is the exact condition every non-vertical, non-horizontal pair of perpendicular lines must satisfy.
Questions
How do you find the slope of a line perpendicular to another line?
Take the original slope, turn it into a reciprocal, and reverse its sign — that gives the perpendicular slope. A slope of 2 becomes −0.5; a slope of −3⁄4 becomes 4⁄3. The two slopes always multiply to exactly −1, which is the algebraic test for a right angle between two lines.
Why is a perpendicular slope the negative reciprocal, not just the reciprocal?
Rotating a line 90° swaps its rise and run and flips one sign, turning a direction vector of (1, slope) into (−slope, 1). Read back as a slope, that ratio is the negative reciprocal of the original — not just its reciprocal. Using only the plain reciprocal, with no sign change, rotates the line the wrong way and does not produce a right angle in general.
How is a perpendicular line different from a parallel line through the same point?
A parallel line keeps the exact same slope as the original — no change at all — so the two lines never meet. A perpendicular line instead uses the negative reciprocal of that slope, so the two lines cross once, at a right angle. Same starting slope, opposite operations applied to it.
What happens when the original slope is 0?
A horizontal line, m = 0, has a perpendicular partner that is perfectly vertical, and vertical lines have no defined slope — you cannot divide by zero to get −1 ⁄ 0. This sheet flags that case instead of returning a number, since a vertical line also cannot be written in y = mx + b form.
How do you find the y-intercept once you have the perpendicular slope?
Substitute the shared point (x₁, y₁) and the new slope into b = y₁ − m_perp·x₁. For slope 2 through (4, 3), the perpendicular slope is −0.5 and b = 3 − (−0.5 × 4) = 3 + 2 = 5, giving the full equation y = −0.5x + 5.
Can two perpendicular lines share the same y-intercept?
Yes — any two slopes satisfying m₁ × m₂ = −1 will meet at a right angle wherever they cross, including at a shared intercept, for example y = 2x + 5 and y = −0.5x + 5. Both pass through (0, 5); one climbs steeply, the other descends gently, and they meet there squarely.