Distance between centers of two intersecting circles if the radii and common chord length is given

Given are two circles, with given radii, which intersect each other and have a common chord. The length of the common chord is given. The task is to find the distance between the center of the two circles.

Examples:

Input:  r1 = 24, r2 = 37, x = 40
Output: 44

Input: r1 = 14, r2 = 7, x = 10
Output: 17

Approach:

• let the length of common chord AB = x
• Let the radius of the circle with center O is OA = r2
• Radius of circle with center P is AP = r1
• From the figure, OP is perpendicular AB
  AC = CB
  AC = x/2 (Since AB = x)
• In triangle ACP,
  AP^2 = PC^2+ AC^2 [By Pythagoras theorem]
  r1^2 = PC^2 + (x/2)^2
  PC^2 = r1^2 – x^2/4
• Consider triangle ACO
  r2^2 = OC^2+ AC^2[By Pythagoras theorem]
  r2^2 = OC^2+ (x/2)^2
  OC^2 = r2^2 – x^2/4
• From the figure, OP = OC + PC
  OP = √( r1^2 – x^2/4 ) + √(r2^2 – x^2/4)

  Distance between the centers = sqrt((radius of one circle)^2 – (half of the length of the common chord )^2) + sqrt((radius of the second circle)^2 – (half of the length of the common chord )^2)

  Below is the implementation of the above approach:
  

  C++

  
  

  
  
  

  // C++ program to find // the distance between centers // of two intersecting circles // if the radii and common chord length is given    #include <bits/stdc++.h> using namespace std;    void distcenter(int r1, int r2, int x) {     int z = sqrt((r1 * r1)                  - (x / 2 * x / 2))             + sqrt((r2 * r2)                    - (x / 2 * x / 2));        cout << "distance between the"          << " centers is "          << z << endl; }    // Driver code int main() {     int r1 = 24, r2 = 37, x = 40;     distcenter(r1, r2, x);     return 0; }

  Java

  
  

  
  
  

  // Java program to find // the distance between centers // of two intersecting circles // if the radii and common chord length is given import java.lang.Math;  import java.io.*;    class GFG {        static double distcenter(int r1, int r2, int x) {     double z = (Math.sqrt((r1 * r1)                 - (x / 2 * x / 2)))             + (Math.sqrt((r2 * r2)                 - (x / 2 * x / 2)));        System.out.println ("distance between the" +                         " centers is "+ (int)z );     return 0; }    // Driver code public static void main (String[] args)  {     int r1 = 24, r2 = 37, x = 40;     distcenter(r1, r2, x); } }    // This code is contributed by jit_t.

  Python3

  
  

  
  
  

  # Python program to find # the distance between centers # of two intersecting circles # if the radii and common chord length is given    def distcenter(r1, r2, x):     z = (((r1 * r1) - (x / 2 * x / 2))**(1/2)) +\     (((r2 * r2)- (x / 2 * x / 2))**(1/2));        print("distance between thecenters is ",end="");     print(int(z));    # Driver code r1 = 24; r2 = 37; x = 40; distcenter(r1, r2, x);    # This code has been contributed by 29AjayKumar

  C#

  
  

  
  
  

  // C# program to find // the distance between centers // of two intersecting circles // if the radii and common chord length is given using System;    class GFG {            static double distcenter(int r1, int r2, int x) {     double z = (Math.Sqrt((r1 * r1)                 - (x / 2 * x / 2)))             + (Math.Sqrt((r2 * r2)                 - (x / 2 * x / 2)));        Console.WriteLine("distance between the" +                         " centers is "+ (int)z );     return 0; }    // Driver code static public void Main () {     int r1 = 24, r2 = 37, x = 40;     distcenter(r1, r2, x); } }    // This code is contributed by jit_t

  Output:

  distance between the centers is 44
  

  
  
  
  
  
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