Section formula (Point that divides a line in given ratio)
Given two coordinates (x1, y1) and (x2, y2), and m and n, find the co-ordinates that divides the line joining (x1, y1) and (x2, y2) in the ratio m : n
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Examples:Â
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Input : x1 = 1, y1 = 0, x2 = 2 y2 = 5,
m = 1, n = 1
Output : (1.5, 2.5)
Explanation: co-ordinates (1.5, 2.5)
divides the line in ratio 1 : 1
Input : x1 = 2, y1 = 4, x2 = 4, y2 = 6,
m = 2, n = 3
Output : (2.8, 4.8)
Explanation: (2.8, 4.8) divides the line
in the ratio 2:3
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The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m : n
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C++
#include <iostream>
using namespace std;
void section( double x1, double x2, double y1,
double y2, double m, double n)
{
double x = ((n * x1) + (m * x2)) /
(m + n);
double y = ((n * y1) + (m * y2)) /
(m + n);
cout << "(" << x << ", " ;
cout << y << ")" << endl;
}
int main()
{
double x1 = 2, x2 = 4, y1 = 4,
y2 = 6, m = 2, n = 3;
section(x1, x2, y1, y2, m, n);
return 0;
}
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Java
import java.io.*;
class sections {
static void section( double x1, double x2,
double y1, double y2,
double m, double n)
{
double x = ((n * x1) + (m * x2)) /
(m + n);
double y = ((n * y1) + (m * y2)) /
(m + n);
System.out.println( "(" + x + ", " + y + ")" );
}
public static void main(String[] args)
{
double x1 = 2 , x2 = 4 , y1 = 4 ,
y2 = 6 , m = 2 , n = 3 ;
section(x1, x2, y1, y2, m, n);
}
}
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Python
def section(x1, x2, y1, y2, m, n):
x = ( float )((n * x1) + (m * x2)) / (m + n)
y = ( float )((n * y1) + (m * y2)) / (m + n)
print (x, y)
x1 = 2
x2 = 4
y1 = 4
y2 = 6
m = 2
n = 3
section(x1, x2, y1, y2, m, n)
|
C#
using System;
class GFG {
static void section( double x1, double x2,
double y1, double y2,
double m, double n)
{
double x = ((n * x1) + (m * x2)) /
(m + n);
double y = ((n * y1) + (m * y2)) /
(m + n);
Console.WriteLine( "(" + x + ", " + y + ")" );
}
public static void Main()
{
double x1 = 2, x2 = 4, y1 = 4,
y2 = 6, m = 2, n = 3;
section(x1, x2, y1, y2, m, n);
}
}
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Javascript
<script>
function section(x1, x2, y1, y2, m, n)
{
let x = ((n * x1) + (m * x2)) /
(m + n);
let y = ((n * y1) + (m * y2)) /
(m + n);
document.write( "(" + x + ", " + y + ")" );
}
let x1 = 2, x2 = 4, y1 = 4,
y2 = 6, m = 2, n = 3;
section(x1, x2, y1, y2, m, n)
</script>
|
PHP
<?php
function section( $x1 , $x2 , $y1 ,
$y2 , $m , $n )
{
$x = (( $n * $x1 ) + ( $m * $x2 ))
/ ( $m + $n );
$y = (( $n * $y1 ) + ( $m * $y2 ))
/ ( $m + $n );
echo ( "(" . $x . ", " );
echo ( $y . ")" );
}
$x1 = 2; $x2 = 4; $y1 = 4;
$y2 = 6; $m = 2; $n = 3;
section( $x1 , $x2 , $y1 , $y2 , $m , $n );
?>
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Output:Â
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(2.8, 4.8)
Time Complexity: O(1)Â
Auxiliary Space: O(1)
How does this work?Â
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From our diagram, we can see,
PS = x – x1 and RT = x2 – x
We are given,
PR/QR = m/n
Using similarity, we can write
RS/QT = PS/RT = PR/QR
Therefore, we can write
PS/RR = m/n
(x - x1) / (x2 - x) = m/n
From above, we get
x = (mx2 + nx1) / (m + n)
Similarly, we can solve for y.
References:Â
http://doubleroot.in/lessons/coordinate-geometry-basics/section-formula/#.WjYXQvbhU8o
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Last Updated :
28 Jan, 2024
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