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

  • Last Updated : 05 Nov, 2021

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:

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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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