# 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 centres 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 centre O is OA = r2
• Radius of circle with centre 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 centres = 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 ` `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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Improved By : jit_t, 29AjayKumar

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