Perpendicular Line Calculator

Perpendicular lines intersect at a right angle of 90 degrees. For two nonvertical lines, their slopes are negative reciprocals: if the original slope is m, a perpendicular line has slope -1/m. Multiplying those slopes gives -1. This relationship turns a geometric condition into a direct algebraic calculation. The calculator combines that slope rule with a point on the desired line. After finding the perpendicular slope, it uses point-slope form y - y₁ = m perpendicular × (x - x₁). Rearranging produces slope-intercept form y = mx + b, where b = y₁ - mx₁. For an original slope of 2 and point (4, 3), the perpendicular slope is -0.5 and the intercept is 5, so the resulting equation is y = -0.5x + 5. Signs matter when taking a negative reciprocal. Start by reversing the numerator and denominator, then change the sign. The negative reciprocal of 2, understood as 2/1, is -1/2. The negative reciprocal of -1/4 is positive 4. A quick check is to multiply the original and perpendicular slopes; the product should be -1 whenever both slopes are finite. Horizontal and vertical lines are the special case. A horizontal line has slope zero, and division by zero cannot produce an ordinary numeric perpendicular slope. Its perpendicular is a vertical line through the supplied point, written x = x₁. Conversely, a vertical original line would have undefined slope and its perpendicular would be horizontal. Because this calculator accepts a numeric original slope, use zero to represent a horizontal original line. Perpendicular equations are useful in analytic geometry, construction layouts, normal vectors, shortest-distance problems, coordinate proofs, and tangent-normal analysis. The selected point belongs to the new line; it does not need to lie on the original line unless the problem specifically asks for a perpendicular through an intersection point. Substituting the point into the displayed equation provides a useful verification. Results are shown with decimal values rounded to six places when needed. Fractions can be entered as decimal equivalents. Keep enough precision when a problem requires an exact or highly accurate answer, and convert the displayed decimal back to a fraction if appropriate. The calculator focuses on the line equation and slope relationship, giving you a fast way to verify hand calculations without hiding the underlying negative-reciprocal rule.