Length of the direct common tangent between two externally touching circles

Given two circles, of given radii, that touch each other externally. The task is to find the length of the direct common tangent between the circles.

Examples:

Input: r1 = 5, r2 = 9
Output: 13.4164

Input: r1 = 11, r2 = 13
Output: 23.9165

Approach

  • Let the radii be r1 & r2 respectively.
  • Draw a line OR parallel to PQ
  • angle OPQ = 90 deg
    angle O’QP = 90 deg

    { line joining the centre of the circle to the point of contact makes an angle of 90 degrees with the tangent }
  • angle OPQ + angle O’QP = 180
    OP || QR
  • Since opposite sides are parallel and interior angles are 90, therefore OPQR is a rectangle.
  • So OP = QR = r1 and PQ = OR = r1+r2
  • In triangle OO’R

    angle ORO’ = 90
    By Pythagoras theorem

    OR^2 + O’R^2 = OO’^2
    OO’^2 = (r1+r2)^2 + (r1-r2)^2

  • So, OO’ = 2√(r1*r2)
      length of the common tangent is = 2sqrt(r1*r2)

Below is the implementation of the above approach:

C++

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// C++ program to find the length of the direct
// common tangent between two circles
// which externally touch each other
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to find the length
// of the direct common tangent
void lengtang(double r1, double r2)
{
    cout << "The length of the "
         << "direct common tangent is "
         << 2 * sqrt(r1 * r2) << endl;
}
  
// Driver code
int main()
{
    double r1 = 5, r2 = 9;
    lengtang(r1, r2);
    return 0;
}

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Java

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// Java program to find the length of the direct 
// common tangent between two circles 
// which externally touch each other 
class GFG 
{
  
    // Function to find the length  
    // of the direct common tangent 
    static void lengtang(double r1, double r2) 
    {
        System.out.println("The length of the "
                + "direct common tangent is "
                + (2 * Math.sqrt(r1 * r2)));
    }
  
    // Driver code 
    public static void main(String[] args)
    {
        double r1 = 5, r2 = 9;
        lengtang(r1, r2);
    }
}
  
// This code contributed by Rajput-Ji

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Python3

# Python3 program to find the length
# of the direct common tangent
# between two circles which
# externally touch each other

# Function to find the length
# of the direct common tangent
def lengtang(r1, r2):
print(“The length of the direct”,
“common tangent is”,
2 * (r1 * r2)**(1 / 2));

# Driver code
r1 = 5; r2 = 9;
lengtang(r1, r2);

# This code contributed
# by PrinciRaj1992

C#

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// C# program to find the length of the direct 
// common tangent between two circles 
// which externally touch each other
using System;
  
class GFG
{
    // Function to find the length 
    // of the direct common tangent 
    static void lengtang(double r1, double r2) 
    {
        Console.WriteLine("The length of the "
                + "direct common tangent is "
                + (2 * Math.Sqrt(r1 * r2)));
    }
      
    // Driver code 
    static public void Main ()
    {
        double r1 = 5, r2 = 9;
        lengtang(r1, r2);
    }
}
  
// This code contributed by ajit.

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PHP

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<?php
// PHP program to find the length of the direct 
// common tangent between two circles 
// which externally touch each other 
  
// Function to find the length 
// of the direct common tangent 
function lengtang($r1, $r2
    echo "The length of the "
        , "direct common tangent is "
        , 2 * sqrt($r1 * $r2) ; 
  
// Driver code 
$r1 = 5; $r2 = 9; 
lengtang($r1, $r2); 
  
// This code is contributed by AnkitRai01
  
?>

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Output:

The length of the direct common tangent is 13.4164


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