Find the length of the median of a Triangle if length of sides are given

Given the length of all three sides of a triangle as a, b and c. The task is to calculate the length of the median of the triangle.
 

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. 
 

Examples: 
 

Input: a = 8, b = 10, c = 13 
Output: 10.89
Input: a = 4, b = 3, c = 5 
Output: 3.61 
 

 



Approach: The idea is to use Apollonius’s theorem to solve this problem.
 

Apollonius’s Theorem states that “the sum of the squares of any two sides of a triangle equals twice the square on half the third side and twice the square on the median bisecting the third side”.
 

From the above figure, According to Apollonius’s Theorm we have: 
 

b^{2} + c^{2} = 2*(a^{2}+m^{2})

where a, b, and c are the length of sides of the triangle 
and m is the length of median of the triangle on side 2*a 
 

Therefore, the length of the median of a triangle from the above equation is given by: 
 

m = \frac{1}{2} \sqrt{2b^{2} + 2c^{2} - a^{2}}

Below is the implementation of the above approach: 
 

C++

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// C++ program to find the length of the 
// median using sides of the triangle
#include<bits/stdc++.h>
using namespace std;
  
// Function to return the length of
// the median using sides of triangle
float median(int a, int b, int c)
{
    float n = sqrt(2 * b * b + 
                   2 * c * c - a * a) / 2;
    return n;
}
  
// Driver code
int main()
{
    int a, b, c;
    a = 4;
    b = 3;
    c = 5;
  
    // Function call
    float ans = median(a, b, c); 
  
    // Print final answer with 2 
    // digits after decimal
    cout << fixed << setprecision(2) << ans;
    return 0;
}
  
// This code is contributed by himanshu77

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Java

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// Java program to find the length of the 
// median using sides of the triangle 
import java.util.*; 
  
class GFG{
      
// Function to return the length of 
// the median using sides of triangle 
public static float median(int a, int b, int c) 
    float n = (float)(Math.sqrt(2 * b * b + 
                                2 * c * c - 
                                a * a) / 2); 
    return n; 
  
// Driver code
public static void main(String[] args)
{
    int a, b, c; 
    a = 4
    b = 3
    c = 5
  
    // Function call 
    float ans = median(a, b, c); 
  
    // Print final answer with 2 
    // digits after decimal 
    System.out.println(String.format("%.2f", ans)); 
}
}
  
// This code is contributed by divyeshrabadiya07

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Python3

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# Python3 implementation to Find the 
# length of the median using sides
# of the triangle
  
import math
  
# Function to return the length of 
# the median using sides of triangle. 
def median(a, b, c): 
   
    n = (1 / 2)*math.sqrt(2*(b**2
   + 2*(c**2)
 - a**2)  
  
    return
  
# Driver Code 
a = 4
b = 3
c = 5
  
# Function Call
ans = median(a, b, c)
  
# Print the final answer
print(round(ans, 2))

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Output: 

3.61

 

Time Complexity: O(1) 
Space Complexity: O(1) 
 

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