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Length of the direct common tangent between two externally touching circles

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  • Last Updated : 07 Jun, 2022

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++




// 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;
}

Java




// 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

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#




// 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.

PHP




<?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
 
?>

Javascript




<script>
 
// javascript 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)
{
    document.write("The length of the "
            + "direct common tangent is "
            + (2 * Math.sqrt(r1 * r2)).toFixed(5));
}
 
// Driver code
var r1 = 5, r2 = 9;
lengtang(r1, r2);
 
// This code contributed by Princi Singh
 
</script>

Output: 

The length of the direct common tangent is 13.4164

 

Time Complexity: O(1)

Auxiliary Space:  O(1)


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