# Find foot of perpendicular from a point in 2 D plane to a Line

Given a point P in 2-D plane and equation of a line, the task is to find the foot of the perpendicular from P to the line.

**Note**: Equation of line is in form ax+by+c=0.

**Examples:**

Input :P=(1, 0), a = -1, b = 1, c = 0Output :Q = (0.5, 0.5) The foot of perpendicular from point (1, 0) to line-x + y = 0is (0.5, 0.5)Input :P=(3, 3), a = 0, b = 1, c = -2Output :Q = (3, 2) The foot of perpendicular from point (3, 3) to liney-2 = 0is (3, 2)

Since equation of the line is given to be of the form ** ax + by + c = 0**. Equation of line passing through P and is perpendicular to line. Therefore equation of line passing through P and Q becomes

**. Also P passes through line passing through P and Q, so we put coordinate of P in the above equation:**

*ay – bx + d = 0*ay1 - bx1 + d = 0 or, d = bx1 - ay1

Also, Q is the intersection of the given line and the line passing through P and Q. So we can find the solution of:

ax + by + c = 0and,ay - bx + (bx1-ay1) = 0

Since a, b, c, d all are known we can find x and y here as:

Below is the implementataion of the above approach:

## C++

`// C++ program for implementation of ` `// the above approach ` `#include <iostream> ` `using` `namespace` `std; ` ` ` `// Function to find foot of perpendicular from ` `// a point in 2 D plane to a Line ` `pair<` `double` `, ` `double` `> findFoot(` `double` `a, ` `double` `b, ` `double` `c, ` ` ` `double` `x1, ` `double` `y1) ` `{ ` ` ` `double` `temp = -1 * (a * x1 + b * y1 + c) / (a * a + b * b); ` ` ` `double` `x = temp * a + x1; ` ` ` `double` `y = temp * b + y1; ` ` ` `return` `make_pair(x, y); ` `} ` ` ` `// Driver Code ` `int` `main() ` `{ ` ` ` `// Equation of line is ` ` ` `// ax + by + c = 0 ` ` ` `double` `a = 0.0; ` ` ` `double` `b = 1.0; ` ` ` `double` `c = -2; ` ` ` ` ` `// Coordinates of point p(x1, y1). ` ` ` `double` `x1 = 3.0; ` ` ` `double` `y1 = 3.0; ` ` ` ` ` `pair<` `double` `, ` `double` `> foot = findFoot(a, b, c, x1, y1); ` ` ` `cout << foot.first << ` `" "` `<< foot.second; ` ` ` ` ` `return` `0; ` `} ` |

*chevron_right*

*filter_none*

## Java

`import` `javafx.util.Pair; ` ` ` `// Java program for implementation of ` `// the above approach ` `class` `GFG ` `{ ` ` ` `// Function to find foot of perpendicular from ` `// a point in 2 D plane to a Line ` `static` `Pair<Double, Double> findFoot(` `double` `a, ` `double` `b, ` `double` `c, ` ` ` `double` `x1, ` `double` `y1) ` `{ ` ` ` `double` `temp = -` `1` `* (a * x1 + b * y1 + c) / (a * a + b * b); ` ` ` `double` `x = temp * a + x1; ` ` ` `double` `y = temp * b + y1; ` ` ` `return` `new` `Pair(x, y); ` `} ` ` ` `// Driver Code ` `public` `static` `void` `main(String[] args) ` `{ ` ` ` `// Equation of line is ` ` ` `// ax + by + c = 0 ` ` ` `double` `a = ` `0.0` `; ` ` ` `double` `b = ` `1.0` `; ` ` ` `double` `c = -` `2` `; ` ` ` ` ` `// Coordinates of point p(x1, y1). ` ` ` `double` `x1 = ` `3.0` `; ` ` ` `double` `y1 = ` `3.0` `; ` ` ` ` ` `Pair<Double, Double> foot = findFoot(a, b, c, x1, y1); ` ` ` `System.out.println(foot.getKey() + ` `" "` `+ foot.getValue()); ` ` ` `} ` `} ` ` ` `// This code contributed by Rajput-Ji ` |

*chevron_right*

*filter_none*

## Python3

`# Python3 implementation of the approach ` ` ` `# Function to find foot of perpendicular ` `# from a point in 2 D plane to a Line ` `def` `findFoot(a, b, c, x1, y1): ` ` ` ` ` `temp ` `=` `(` `-` `1` `*` `(a ` `*` `x1 ` `+` `b ` `*` `y1 ` `+` `c) ` `/` `/` ` ` `(a ` `*` `a ` `+` `b ` `*` `b)) ` ` ` `x ` `=` `temp ` `*` `a ` `+` `x1 ` ` ` `y ` `=` `temp ` `*` `b ` `+` `y1 ` ` ` `return` `(x, y) ` ` ` `# Driver Code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `# Equation of line is ` ` ` `# ax + by + c = 0 ` ` ` `a, b, c ` `=` `0.0` `, ` `1.0` `, ` `-` `2` ` ` ` ` `# Coordinates of point p(x1, y1). ` ` ` `x1, y1 ` `=` `3.0` `, ` `3.0` ` ` ` ` `foot ` `=` `findFoot(a, b, c, x1, y1) ` ` ` `print` `(` `int` `(foot[` `0` `]), ` `int` `(foot[` `1` `])) ` ` ` `# This code is contributed ` `# by Rituraj Jain ` |

*chevron_right*

*filter_none*

## C#

`// C# program for implementation of ` `// the above approach ` `using` `System; ` ` ` `class` `GFG ` `{ ` ` ` `// Pair class ` ` ` `public` `class` `Pair ` ` ` `{ ` ` ` `public` `double` `first,second; ` ` ` `public` `Pair(` `double` `a,` `double` `b) ` ` ` `{ ` ` ` `first = a; ` ` ` `second = b; ` ` ` `} ` ` ` `} ` ` ` `// Function to find foot of perpendicular from ` `// a point in 2 D plane to a Line ` `static` `Pair findFoot(` `double` `a, ` `double` `b, ` `double` `c, ` ` ` `double` `x1, ` `double` `y1) ` `{ ` ` ` `double` `temp = -1 * (a * x1 + b * y1 + c) / (a * a + b * b); ` ` ` `double` `x = temp * a + x1; ` ` ` `double` `y = temp * b + y1; ` ` ` `return` `new` `Pair(x, y); ` `} ` ` ` `// Driver Code ` `public` `static` `void` `Main(String []args) ` `{ ` ` ` `// Equation of line is ` ` ` `// ax + by + c = 0 ` ` ` `double` `a = 0.0; ` ` ` `double` `b = 1.0; ` ` ` `double` `c = -2; ` ` ` ` ` `// Coordinates of point p(x1, y1). ` ` ` `double` `x1 = 3.0; ` ` ` `double` `y1 = 3.0; ` ` ` ` ` `Pair foot = findFoot(a, b, c, x1, y1); ` ` ` `Console.WriteLine(foot.first + ` `" "` `+ foot.second); ` ` ` `} ` `} ` ` ` `// This code contributed by Arnab Kundu ` |

*chevron_right*

*filter_none*

## PHP

`<?php ` `// PHP implementation of the approach ` ` ` `// Function to find foot of perpendicular ` `// from a point in 2 D plane to a Line ` `function` `findFoot(` `$a` `, ` `$b` `, ` `$c` `, ` `$x1` `, ` `$y1` `) ` `{ ` ` ` ` ` `$temp` `= ` `floor` `((-1 * (` `$a` `* ` `$x1` `+ ` `$b` `* ` `$y1` `+ ` `$c` `) / ` ` ` `(` `$a` `* ` `$a` `+ ` `$b` `* ` `$b` `))); ` ` ` `$x` `= ` `$temp` `* ` `$a` `+ ` `$x1` `; ` ` ` `$y` `= ` `$temp` `* ` `$b` `+ ` `$y1` `; ` ` ` `return` `array` `(` `$x` `, ` `$y` `); ` ` ` `} ` ` ` `// Driver Code ` ` ` `// Equation of line is ` `// ax + by + c = 0 ` `$a` `= 0.0; ` `$b` `= 1.0 ; ` `$c` `= -2 ; ` ` ` `// Coordinates of point p(x1, y1). ` `$x1` `= 3.0 ; ` `$y1` `= 3.0 ; ` ` ` `$foot` `= findFoot(` `$a` `, ` `$b` `, ` `$c` `, ` `$x1` `, ` `$y1` `); ` `echo` `floor` `(` `$foot` `[0]), ` `" "` `, ` `floor` `(` `$foot` `[1]); ` ` ` `// This code is contributed by Ryuga ` `?> ` |

*chevron_right*

*filter_none*

**Output:**

3 2

## Recommended Posts:

- Find the foot of perpendicular of a point in a 3 D plane
- Perpendicular distance between a point and a Line in 2 D
- Shortest distance between a Line and a Point in a 3-D plane
- Find mirror image of a point in 2-D plane
- Slope of perpendicular to line
- Find the other end point of a line with given one end and mid
- Program to find the mid-point of a line
- Length of the perpendicular bisector of the line joining the centers of two circles
- Find an Integer point on a line segment with given two ends
- Equation of straight line passing through a given point which bisects it into two equal line segments
- Distance between a point and a Plane in 3 D
- Mirror of a point through a 3 D plane
- Check if a line at 45 degree can divide the plane into two equal weight parts
- Number of jump required of given length to reach a point of form (d, 0) from origin in 2D plane
- Reflection of a point about a line in C++

If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.