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Perpendicular Line Calculator

Find the line perpendicular to a given slope that passes through your point.

How it is calculated

A line at 90° to a line of slope m has the negative reciprocal slope, m⊥ = −1 / m. The tool computes that, then places the new line through your point by solving for the intercept: b = y − m⊥ × x. The two outputs, the perpendicular slope and the y-intercept b, are shown with four decimals, so the line is y = m⊥·x + b. You enter the slope of the original line (not its equation) and the coordinates of the point the new line must pass through. A slope of 0 gives no result: the perpendicular to a horizontal line is vertical, and a vertical line has no slope or y-intercept to display.

Worked example

The original line has slope 4, and the perpendicular must pass through (8, 1). Step 1: the perpendicular slope is −1 / 4 = -0.25. Step 2: b = 1 − (−0.25) × 8 = 1 + 2 = 3. The tool prints a perpendicular slope of -0.2500 and a y-intercept of 3.0000, so the line is y = −0.25x + 3. Check: at x = 8 it gives −2 + 3 = 1.

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Common mistakes

  • Using only the opposite sign (−m) or only the reciprocal (1/m). A perpendicular slope needs both: flip and negate.
  • Entering the slope of a line that is already perpendicular and expecting the original back. The relation works both ways, so the result is the same rule applied again.
  • Entering 0 for a horizontal line. The perpendicular is vertical, x = constant, which this tool cannot display.
  • Putting the original line's intercept into the point fields. The point is where the new line must pass, not where the old one crosses the y-axis.
  • Forgetting the sign in b = y − m⊥ × x; with a negative m⊥ the two minus signs add.

About this calculator

Two lines are perpendicular when their slopes are negative reciprocals of each other — if a given line has slope m, any line perpendicular to it has slope −1/m, a direct result of the geometric fact that perpendicular lines meet at a 90-degree angle. Once that perpendicular slope is known, plugging in a specific point through the point-slope form yields the full equation of the perpendicular line, including its y-intercept.

This relationship is a core tool in coordinate geometry, used to construct perpendicular bisectors of segments, find the shortest distance from a point to a line, determine altitudes of triangles, and check whether shapes have right angles — skills that carry directly into CAD drafting, computer graphics, physics (like resolving forces into perpendicular components), and standardized math tests.

This calculator takes a given slope and a point the perpendicular line must pass through, then returns the perpendicular slope and the y-intercept of that line, rounded to four decimal places; together they give its equation, y = (slope)x + (intercept). The given slope cannot be zero, since the perpendicular line would then be vertical.

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