Given the height 
of an isosceles triangle and two integers 
, 
. The task is to find the height 
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 heightfloat 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 codeint 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:mimport java.io.*;class GFG {// Function to return the heightstatic 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:mfrom math import sqrt# Function to return the heightdef 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 codeif __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:musing System;class GFG {// Function to return the heightstatic 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 codepublic 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 heightfunction 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> |
6.54654
Time Complexity: O(log(n)) as it is using inbuilt sqrt function
Auxiliary Space: O(1)
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