Perpendicular Distance Calculator
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About Perpendicular Distance
The perpendicular distance from a point to a line is the shortest distance between the point and the line. It is measured along a line that is perpendicular to the given line and passes through the given point.
Perpendicular Distance Formulas
From a Point to a Line in Standard Form
For a point $(x_0, y_0)$ and a line $Ax + By + C = 0$, the perpendicular distance is:
$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$
From a Point to a Line in Slope-Intercept Form
For a point $(x_0, y_0)$ and a line $y = mx + b$, the perpendicular distance is:
$d = \frac{|y_0 - mx_0 - b|}{\sqrt{1 + m^2}}$
From a Point to a Line Defined by Two Points
For a point $P(x_0, y_0)$ and a line passing through points $A(x_1, y_1)$ and $B(x_2, y_2)$, the perpendicular distance is:
$d = \frac{|(y_2 - y_1)x_0 - (x_2 - x_1)y_0 + x_2y_1 - y_2x_1|}{\sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}}$
Applications
- Finding the shortest distance from a point to a line
- Determining if a point lies on a line (distance = 0)
- Calculating the height of a triangle
- Finding the distance between parallel lines
- Computer graphics and computational geometry