将等腰三角形分为两部分，面积比为n：m

给定高度$H$等腰三角形和两个整数$n$ ， $m$ 。任务是找到高度$h$从三角形的顶部开始，这样，如果我们从顶部平行于底部在高度h处进行切割，则三角形必须分成两部分，其面积比等于n：m 。
例子：

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

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

首先，在继续之前，让我们提及等腰三角形的一些基本属性。
假设▲ABC是等腰三角形，AB = AC，BC是三角形的边和底边不相等。如果D是BC的中点，则AD是我们的高度H。此外，如果我们画一条与BC的平行线，分别在E和F点切开AB和AC，而G是EF的中点，则▲AEG类似于▲ABD，▲AFG类似于▲ACD，▲AEF类似于▲ABC，通过使用相似三角形的属性，我们可以得出以下几点：
AE / AB = AG / AD = EG / BD = EF / BC = AF / AC-(i)

根据问题的要求，我们必须找出高度h，以使▲AEF面积与梯形EFCB面积之比= 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)) 
 

下面是上述方法的实现：

C++

// C++ program, to find height h
// which divide isosceles triangle
// into ratio n:m
#include 
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

Javascript

 
// 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.
 

输出：

6.54654