Divide an isosceles triangle in two parts with ratio of areas as n:m

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Given the height $H$ of an isosceles triangle and two integers $n$ , $m$ . The task is to find the height $h$ from top of the triangle such that if we make a cut at height h from top parallel to base then the triangle must be divided into two parts with the ratio of their areas equal to n:m.
Examples:

Input : H = 4, n = 1, m = 1
Output : 2.82843

Input : H = 4, n = 1, m = 0
Output : 4

First of all before proceeding let us mention some of the basic properties of an isosceles triangle.
Let ?ABC is an isosceles triangle with AB = AC and BC being unequal side and base of the triangle. If D is mid-point of BC, then AD is our height H. Also, if we draw a parallel line to BC which cuts AB and AC at points E and F respectively and G being the midpoint of EF then ?AEG is similar to ?ABD, ?AFG is similar to ?ACD, ?AEF is similar to ?ABC, and by using properties of similar triangles we can conclude the following points:
AE/AB = AG/AD = EG/BD = EF/BC = AF/AC —–(i)

As per problem’s requirement, we have to find out the height h, such that the ratio of the area of ?AEF to the area of trapezium EFCB = n:m.

Let, h is the height of cut from the top of the triangle.
Now, area of ?AEF = 0.5 * AG * EF and area of trapezium EFCB = 0.5 * GD * (EF+BC)
also, ratio of both is n:m.
So, we can say that ratio of area of ?AEF to area of ?ABC is equal to n :(n+m).
=> area of ?AEF / area of ?ABC = n / (n+m)
=> (0.5 * AG * EF) / (0.5 * AD * BC) = n / (n+m)
=> (AG/AD) * (EF/BC) = n / (n+m)
=> (EF/BC) * (EF/BC) = n / (n+m)
=> h² /H² = n / (n+m)
=> h = H*sqrt(n/(n+m))

Below is the implementation of the above approach:

C++

// C++ program, to find height h
// which divide isosceles triangle
// into ratio n:m
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the height
float heightCalculate(int H, int n, int m)
{
    // type cast the n, m into float
    float N = n * 1.0;
    float M = m * 1.0;
    // calculate the height for cut
    float h = H * sqrt(N / (N + M));
    return h;
}
 
// Driver code
int main()
{
    int H = 10, n = 3, m = 4;
    cout << heightCalculate(H, n, m);
    return 0;
}

Java

// Java program, to find height h
// which divide isosceles triangle
// into ratio n:m
 
import java.io.*;
 
class GFG {
 
 
// Function to return the height
static float heightCalculate(int H, int n, int m)
{
    // type cast the n, m into float
    float N = (float)(n * 1.0);
    float M = (float)(m * 1.0);
    // calculate the height for cut
    float h = H *(float) Math.sqrt(N / (N + M));
    return h;
}
 
// Driver code
 
 
    public static void main (String[] args) {
            int H = 10, n = 3, m = 4;
    System.out.print( heightCalculate(H, n, m));
    }
}

Python3

# Python 3 program, to find height 
# h which divide isosceles triangle
# into ratio n:m
from math import sqrt
 
# Function to return the height
def heightCalculate(H, n, m):
     
    # type cast the n, m into float
    N = n * 1.0
    M = m * 1.0
     
    # calculate the height for cut
    h = H * sqrt(N / (N + M))
    return h
 
# Driver code
if __name__ == '__main__':
    H = 10
    n = 3
    m = 4
    print("{0:.6}" . 
    format(heightCalculate(H, n, m)));
     
# This code is contributed 
# by Surendra_Gangwar

C#

// C# program, to find height h
// which divide isosceles triangle
// into ratio n:m
using System;
 
class GFG 
{
 
// Function to return the height
static float heightCalculate(int H,
                             int n, int m)
{
    // type cast the n, m into float
    float N = (float)(n * 1.0);
    float M = (float)(m * 1.0);
     
    // calculate the height for cut
    float h = H * (float) Math.Sqrt(N / (N + M));
    return h;
}
 
// Driver code
public static void Main ()
{
    int H = 10, n = 3, m = 4;
    Console.WriteLine(heightCalculate(H, n, m));
}
}
 
// This code is contributed
// by inder_verma

PHP

<?php
// PHP program, to find height h
// which divide isosceles triangle
// into ratio n:m
 
// Function to return the height
function heightCalculate($H, $n, $m)
{
    // type cast the n, m into float
    $N = $n * 1.0;
    $M = $m * 1.0;
     
    // calculate the height for cut
    $h = $H * sqrt($N / ($N + $M));
    return $h;
}
 
// Driver code
$H = 10; $n = 3; $m = 4;
echo heightCalculate($H, $n, $m);
 
// This code is contributed
// by anuj_67
?>

Javascript

<script>
 
// JavaScript program, to find height h 
// which divide isosceles triangle 
// into ratio n:m 
 
// Function to return the height 
function heightCalculate(H, n, m) 
{ 
    // type cast the n, m into float 
    let N = n * 1.0; 
    let M = m * 1.0; 
    // calculate the height for cut 
    let h = H * Math.sqrt(N / (N + M)); 
    return h; 
} 
 
// Driver code 
 
    let H = 10, n = 3, m = 4; 
    document.write(heightCalculate(H, n, m)); 
 
 
// This code is contributed by Surbhi Tyagi.
 
</script>

Output

6.54654

Time Complexity: O(log(n)) as it is using inbuilt sqrt function
Auxiliary Space: O(1)

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