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Find the coordinates of a triangle whose Area = (S / 2)

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  • Last Updated : 01 Mar, 2022

Given an integer S, the task is to find the coordinates of a triangle whose area is (S / 2).

Examples: 

Input: S = 4 
Output: 
(0, 0) 
(1000000000, 1) 
(999999996, 1)

Input: S = 15 
Output: 
(0, 0) 
(1000000000, 1) 
(999999985, 1) 
 

Approach:  

  • It is know than the area of the triangle whose coordinates are (X1, Y1), (X2, Y2) and (X3, Y3) is given by A = ((X1 * Y2) + (X2 * Y3) + (X3 * Y1) – (X1 * Y3) – (X2 * Y1) – (X3 * Y2)) / 2.
  • Now fixing (X1, Y1) to (0, 0) gives A = ((X2 * Y3) – (X3 * Y2)) / 2.
  • It is given that A = S / 2 which implies S = (X2 * Y3) – (X3 * Y2).
  • Now fix (X2, Y2) to (109, 1) and the equation becomes S = 109 * Y3 – X3 which can be solved by taking an integer value of a variable that given the integer value for the other variable.

Below is the implementation of the above approach:  

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
const long MAX = 1000000000;
 
// Function to find the triangle
// with area = (S / 2)
void findTriangle(long S)
{
 
    // Fix the two pairs of coordinates
    long X1 = 0, Y1 = 0;
    long X2 = MAX, Y2 = 1;
 
    // Find (X3, Y3) with integer coordinates
    long X3 = (MAX - S % MAX) % MAX;
    long Y3 = (S + X3) / MAX;
 
    cout << "(" << X1 << ", " << Y1 << ")\n";
    cout << "(" << X2 << ", " << Y2 << ")\n";
    cout << "(" << X3 << ", " << Y3 << ")";
}
 
// Driver code
int main()
{
 
    long S = 4;
 
    findTriangle(S);
 
    return 0;
}

Java




// Java implementation of the approach
class GFG
{
    static final long MAX = 1000000000;
     
    // Function to find the triangle
    // with area = (S / 2)
    static void findTriangle(long S)
    {
     
        // Fix the two pairs of coordinates
        long X1 = 0, Y1 = 0;
        long X2 = MAX, Y2 = 1;
     
        // Find (X3, Y3) with integer coordinates
        long X3 = (MAX - S % MAX) % MAX;
        long Y3 = (S + X3) / MAX;
     
        System.out.println("(" + X1 +
                           ", " + Y1 + ")");
        System.out.println("(" + X2 +
                           ", " + Y2 + ")");
        System.out.println("(" + X3 +
                           ", " + Y3 + ")");
    }
     
    // Driver code
    public static void main (String[] args)
    {
        long S = 4;
     
        findTriangle(S);
    }
}
 
// This code is contributed by AnkitRai01

Python3




# Python3 implementation of the approach
MAX = 1000000000;
 
# Function to find the triangle
# with area = (S / 2)
def findTriangle(S) :
 
    # Fix the two pairs of coordinates
    X1 = 0; Y1 = 0;
    X2 = MAX; Y2 = 1;
 
    # Find (X3, Y3) with integer coordinates
    X3 = (MAX - S % MAX) % MAX;
    Y3 = (S + X3) / MAX;
 
    print("(", X1, ",", Y1, ")");
    print("(", X2, ",", Y2, ")");
    print("(", X3, ",", Y3, ")");
 
# Driver code
if __name__ == "__main__" :
 
    S = 4;
 
    findTriangle(S);
 
# This code is contributed by kanugargng

C#




// C# implementation of the above approach
using System;
     
class GFG
{
    static readonly long MAX = 1000000000;
     
    // Function to find the triangle
    // with area = (S / 2)
    static void findTriangle(long S)
    {
     
        // Fix the two pairs of coordinates
        long X1 = 0, Y1 = 0;
        long X2 = MAX, Y2 = 1;
     
        // Find (X3, Y3) with integer coordinates
        long X3 = (MAX - S % MAX) % MAX;
        long Y3 = (S + X3) / MAX;
     
        Console.WriteLine("(" + X1 +
                         ", " + Y1 + ")");
        Console.WriteLine("(" + X2 +
                         ", " + Y2 + ")");
        Console.WriteLine("(" + X3 +
                         ", " + Y3 + ")");
    }
     
    // Driver code
    public static void Main (String[] args)
    {
        long S = 4;
     
        findTriangle(S);
    }
}
 
// This code is contributed by PrinciRaj1992

Javascript




<script>
 
// Javascript implementation of the approach
 
let MAX = 1000000000;
 
// Function to find the triangle
// with area = (S / 2)
function findTriangle( S)
{
 
    // Fix the two pairs of coordinates
    let X1 = 0, Y1 = 0;
    let X2 = MAX, Y2 = 1;
 
    // Find (X3, Y3) with integer coordinates
    let X3 = (MAX - S % MAX) % MAX;
    let Y3 = (S + X3) / MAX;
 
    document.write( "(" + X1 + ", " + Y1 + ")<br/>");
    document.write( "(" + X2 + ", " + Y2 + ")<br/>");
    document.write( "(" + X3 + ", " + Y3 + ")<br/>")
    }
 
// Driver code
 
    let S = 4;
 
    findTriangle(S);
 
// This code contributed by aashish1995
 
</script>

Output: 

(0, 0)
(1000000000, 1)
(999999996, 1)

 

Time Complexity: O(1)

Auxiliary Space: O(1)
 


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