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# Minimum length of the shortest path of a triangle

Given N points on the plane, ( X1, Y1 ), ( X2, Y2 ), ( X3, Y3 ), ……, ( XN, YN ). The task is to calculate the minimum length of the shorter side of the triangle. and the path or points to place an isosceles triangle with any two sides of the triangle on the coordinate axis ( X axis and Y axis ) to cover all points.
Note: A point is covered if it lies inside the triangle or on the side of the triangle.
Examples:

Input: (1, 3), (1, 1), (2, 1), (2, 2)
Output: Length -> 4 , Path -> ( 1, 4 ) and ( 4, 1 )
Input: (1, 2), (1, 1), (2, 1)
Output: Length -> 3 , Path -> ( 1, 3 ) and ( 3, 1 ) In the first example, the minimum length of the shortest path is equal to the maximum sum of the points, which is 1+3 or 2+2. So the path which will cover all the points is (1, 4) and (4, 1) on the coordinate axis.
Below is the step by step algorithm to solve this problem:

1. Initialize ‘N’ points on a plane.
2. Traverse through each point and find the sum of each point and store it in a variable ‘answer’.
3. Replace the next maximum sum of the points with the previous sum.
4. Then you will get the path on a coordinate axis ( 1, answer ) and ( answer, 1 ) which will cover all the points of isosceles triangle.

Below is the implementation of above algorithm:

## C++

 `// C++ program to illustrate``// the above problem``#include ``using` `namespace` `std;``#define ll long long` `// function to get the minimum length of``// the shorter side of the triangle``void` `shortestLength(``int` `n, ``int` `x[], ``int` `y[])``{``        ``int` `answer = 0;` `    ``// traversing through each points on the plane``    ``int` `i = 0;``    ``while` `(n--) {``        ``// if sum of a points is greater than the``        ``// previous one, the maximum gets replaced``        ``if` `(x[i] + y[i] > answer)``            ``answer = x[i] + y[i];` `        ``i++;``    ``}` `    ``// print the length``    ``cout << ``"Length -> "` `<< answer << endl;``    ``cout << ``"Path -> "``         ``<< ``"( 1, "` `<< answer << ``" )"``         ``<< ``"and ( "` `<< answer << ``", 1 )"``;``}` `// Driver code``int` `main()``{``    ``// initialize the number of points``    ``int` `n = 4;` `    ``// points on the plane``    ``int` `x[n] = { 1, 4, 2, 1 };``    ``int` `y[n] = { 4, 1, 1, 2 };` `    ``shortestLength(n, x, y);` `    ``return` `0;``}`

## Java

 `// Java program to illustrate``// the above problem``class` `GFG``{` `// function to get the minimum length of``// the shorter side of the triangle``static` `void` `shortestLength(``int` `n, ``int` `x[],``                           ``int` `y[])``{``    ``int` `answer = ``0``;` `    ``// traversing through each``    ``// points on the plane``    ``int` `i = ``0``;``    ``while` `(n != ``0` `&& i < x.length)``    ``{``        ``// if sum of a points is greater``        ``// than the previous one, the``        ``// maximum gets replaced``        ``if` `(x[i] + y[i] > answer)``            ``answer = x[i] + y[i];` `        ``i++;``    ``}` `    ``// print the length``    ``System.out.println(``"Length -> "` `+ answer );``    ``System.out.println(``"Path -> "` `+``                       ``"( 1, "` `+ answer + ``" )"` `+``                       ``"and ( "` `+ answer + ``", 1 )"``);``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``// initialize the number of points``    ``int` `n = ``4``;` `    ``// points on the plane``    ``int` `x[] = ``new` `int``[] { ``1``, ``4``, ``2``, ``1` `};``    ``int` `y[] = ``new` `int``[] { ``4``, ``1``, ``1``, ``2` `};` `    ``shortestLength(n, x, y);``}``}` `// This code is contributed``// by Prerna Saini`

## Python 3

 `# Python 3 program to illustrate``# the above problem` `# function to get the minimum length of``# the shorter side of the triangle``def` `shortestLength(n, x, y):``    ``answer ``=` `0` `    ``# traversing through each``    ``# points on the plane``    ``i ``=` `0``    ``while` `n > ``0``:``        ` `        ``# if sum of a points is greater``        ``# than the previous one, the``        ``# maximum gets replaced``        ``if` `(x[i] ``+` `y[i] > answer):``            ``answer ``=` `x[i] ``+` `y[i]` `        ``i ``+``=` `1``        ``n ``-``=` `1` `    ``# print the length``    ``print``(``"Length -> "``+` `str``(answer))``    ``print``( ``"Path -> "``+``        ``"( 1, "` `+``str``(answer)``+` `" )"``+``        ``"and ( "``+``str``( answer) ``+``", 1 )"``)` `# Driver code``if` `__name__ ``=``=` `"__main__"``:``    ` `    ``# initialize the number of points``    ``n ``=` `4` `    ``# points on the plane``    ``x ``=` `[ ``1``, ``4``, ``2``, ``1` `]``    ``y ``=` `[ ``4``, ``1``, ``1``, ``2` `]``    ``shortestLength(n, x, y)` `# This code is contributed``# by ChitraNayal`

## C#

 `// C# program to illustrate``// the above problem``using` `System;` `class` `GFG``{` `// function to get the minimum``// length of the shorter side``// of the triangle``static` `void` `shortestLength(``int` `n, ``int``[] x,``                           ``int``[] y)``{``    ``int` `answer = 0;` `    ``// traversing through each``    ``// points on the plane``    ``int` `i = 0;``    ``while` `(n != 0 && i < x.Length)``    ``{``        ``// if sum of a points is greater``        ``// than the previous one, the``        ``// maximum gets replaced``        ``if` `(x[i] + y[i] > answer)``            ``answer = x[i] + y[i];` `        ``i++;``    ``}` `    ``// print the length``    ``Console.WriteLine(``"Length -> "` `+ answer);``    ``Console.WriteLine(``"Path -> "` `+``                      ``"( 1, "` `+ answer + ``" )"` `+``                      ``"and ( "` `+ answer + ``", 1 )"``);``}` `// Driver code``static` `public` `void` `Main ()``{``    ` `    ``// initialize the``    ``// number of points``    ``int` `n = 4;` `    ``// points on the plane``    ``int``[] x = ``new` `int``[] { 1, 4, 2, 1 };``    ``int``[] y = ``new` `int``[] { 4, 1, 1, 2 };` `    ``shortestLength(n, x, y);``}``}` `// This code is contributed by Mahadev`

## PHP

 ` ``\$answer``)``            ``\$answer` `= ``\$x``[``\$i``] + ``\$y``[``\$i``];` `        ``\$i``++;``    ``}` `    ``// print the length``    ``echo` `"Length -> "``.``\$answer``.``"\n"``;``    ``echo` `"Path -> "``. ``"( 1, "` `.``\$answer` `.``" )"``.``         ``"and ( "` `.``\$answer` `. ``", 1 )"``;``}` `// Driver code` `// initialize the number of points``\$n` `= 4;` `// points on the plane``\$x` `= ``array``(1, 4, 2, 1 );``\$y` `= ``array``(4, 1, 1, 2 );` `shortestLength(``\$n``, ``\$x``, ``\$y``);` `// This code is contributed``// by ChitraNayal``?>`

## Javascript

 ``

Output:

```Length -> 5
Path -> ( 1, 5 )and ( 5, 1 )```

Time Complexity: O(n) where n is the number of points.
Auxiliary Space: O(1)