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# Find other two sides and angles of a right angle triangle

Given one side of right angle triangle, check if there exists a right angle triangle possible with any other two sides of the triangle. If possible print length of the other two sides and all the angles of the triangle. Examples:

Input : a = 12
Output : Sides are a = 12, b = 35, c = 37
Angles are A = 18.9246, B = 71.0754, C = 90
Explanation: a = 12, b = 35 and c = 37 form right
angle triangle because
12*12 + 35*35 = 37*37
Input : a = 6
Output : Sides are a = 6, b = 8, c = 10
Angles are A = 36.8699, B = 53.1301, C = 90

Approach to check if triangle exists and finding Sides
To solve this problem we first observe the Pythagoras equation. If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
This relationship is represented by the formula:

a*a + b*b = c*c

Case 1: a is an odd number: Given a, find b and c

c2 - b2 = a2
OR
c = (a2 + 1)/2;
b = (a2 - 1)/2;

Above solution works only for case when a is odd, because a2 + 1 is divisible by 2 only for odd a.
Case 2 : a is an even number: When c-b is 2 & c+b is (a2)/2

c-b = 2 & c+b = (a2)/2
Hence,
c = (a2)/4 + 1;
b = (a2)/4 - 1;

This works when a is even.
Approach to find Angles
First find all sides of triangle. Then Applied “SSS” rule that’s means law of cosine: Below is the implementation of the above approach:

## C++

 // C++ program to print all sides and angles of right // angle triangle given one side #include  #include  using namespace std;   #define PI 3.1415926535   // Function to find angle A  // Angle in front of side a double findAnglesA(double a, double b, double c) {     // applied cosine rule     double A = acos((b * b + c * c - a * a) / (2 * b * c));       // convert into degrees     return A * 180 / PI; }   // Function to find angle B  // Angle in front of side b double findAnglesB(double a, double b, double c) {     // applied cosine rule     double B = acos((a * a + c * c - b * b) / (2 * a * c));       // convert into degrees and return     return B * 180 / PI; }   // Function to print all angles  // of the right angled triangle void printAngles(int a, int b, int c) {     double x = (double)a;     double y = (double)b;     double z = (double)c;           // for calculate angle A     double A = findAnglesA(x, y, z);           // for calculate angle B     double B = findAnglesB(x, y, z);           cout << "Angles are A = " << A << ", B = " <<                         B << ", C = " << 90 << endl; }   // Function to find other two sides of the // right angled triangle void printOtherSides(int n)  {        int b,c;           // if n is odd      if (n & 1)      {          // case of n = 1 handled separately          if (n == 1)              cout << -1 << endl;          else         {              b = (n*n-1)/2;              c = (n*n+1)/2;              cout << "Side b = " << b                  << ", Side c = " << c << endl;          }      }      else     {          // case of n = 2 handled separately          if (n == 2)              cout << -1 << endl;          else         {              b = n*n/4-1;              c = n*n/4+1;              cout << "Side b = " << b                  << ", Side c = " << c << endl;          }      }            // Print angles of the triangle     printAngles(n,b,c); }    // Driver Program int main() {     int a = 12;       printOtherSides(a);           return 0; }

## Java

 // Java program to print all sides and angles of right // angle triangle given one side     import java.io.*;   class GFG {      static double  PI = 3.1415926535;   // Function to find angle A  // Angle in front of side a static double findAnglesA(double a, double b, double c) {     // applied cosine rule     double A = Math.acos((b * b + c * c - a * a) / (2 * b * c));       // convert into degrees     return A * 180 / PI; }   // Function to find angle B  // Angle in front of side b static double findAnglesB(double a, double b, double c) {     // applied cosine rule     double B = Math.acos((a * a + c * c - b * b) / (2 * a * c));       // convert into degrees and return     return B * 180 / PI; }   // Function to print all angles  // of the right angled triangle static void printAngles(int a, int b, int c) {     double x = (double)a;     double y = (double)b;     double z = (double)c;           // for calculate angle A     double A = findAnglesA(x, y, z);           // for calculate angle B     double B = findAnglesB(x, y, z);           System.out.println( "Angles are A = " + A + ", B = " +                         B + ", C = " + 90); }   // Function to find other two sides of the // right angled triangle static void printOtherSides(int n)  {      int b=0,c=0;           // if n is odd      if ((n & 1)>0)      {          // case of n = 1 handled separately          if (n == 1)              System.out.println( -1);          else         {              b = (n*n-1)/2;              c = (n*n+1)/2;              System.out.println( "Side b = " + b                  + ", Side c = " + c );          }      }      else     {          // case of n = 2 handled separately          if (n == 2)              System.out.println( -1);          else         {              b = n*n/4-1;              c = n*n/4+1;              System.out.println( "Side b = " + b                  + ", Side c = " + c);          }      }            // Print angles of the triangle     printAngles(n,b,c); }    // Driver Program         public static void main (String[] args) {     int a = 12;       printOtherSides(a);     } }   // This code is contributed  // by inder_verma..

## Python 3

 # Python 3 program to print all  # sides and angles of right  # angle triangle given one side import math   PI = 3.1415926535   # Function to find angle A  # Angle in front of side a def findAnglesA( a, b, c):           # applied cosine rule     A = math.acos((b * b + c * c - a * a) /                               (2 * b * c))       # convert into degrees     return A * 180 / PI   # Function to find angle B  # Angle in front of side b def findAnglesB(a, b, c):       # applied cosine rule     B = math.acos((a * a + c * c - b * b) /                               (2 * a * c))       # convert into degrees      # and return     return B * 180 / PI   # Function to print all angles  # of the right angled triangle def printAngles(a, b, c):       x = a     y = b     z = c           # for calculate angle A     A = findAnglesA(x, y, z)       # for calculate angle B     B = findAnglesB(x, y, z)           print("Angles are A = ", A,            ", B = ", B , ", C = ", "90 ")   # Function to find other two sides  # of the right angled triangle def printOtherSides(n):           # if n is odd      if (n & 1) :                   # case of n = 1 handled          # separately          if (n == 1):              print("-1")          else:                           b = (n * n - 1) // 2             c = (n * n + 1) // 2             print("Side b = ", b,                    " Side c = ", c)           else:                   # case of n = 2 handled          # separately          if (n == 2) :             print("-1")         else:             b = n * n // 4 - 1;             c = n * n // 4 + 1;             print("Side b = " , b,                    ", Side c = " , c)               # Print angles of the triangle     printAngles(n, b, c)    # Driver Code if __name__ == "__main__":     a = 12       printOtherSides(a)   # This code is contributed  # by ChitraNayal

## C#

 // C# program to print all sides  // and angles of right angle  // triangle given one side using System;   class GFG  { static double PI = 3.1415926535;   // Function to find angle A  // Angle in front of side a static double findAnglesA(double a,                            double b, double c) {     // applied cosine rule     double A = Math.Acos((b * b + c *                            c - a * a) /                           (2 * b * c));       // convert into degrees     return A * 180 / PI; }   // Function to find angle B  // Angle in front of side b static double findAnglesB(double a,                            double b, double c) {     // applied cosine rule     double B = Math.Acos((a * a + c *                            c - b * b) /                           (2 * a * c));       // convert into degrees and return     return B * 180 / PI; }   // Function to print all angles  // of the right angled triangle static void printAngles(int a, int b, int c) {     double x = (double)a;     double y = (double)b;     double z = (double)c;           // for calculate angle A     double A = findAnglesA(x, y, z);           // for calculate angle B     double B = findAnglesB(x, y, z);           Console.WriteLine( "Angles are A = " +                             A + ", B = " +                         B + ", C = " + 90); }   // Function to find other two sides  // of the right angled triangle static void printOtherSides(int n)  {      int b = 0, c = 0;           // if n is odd      if ((n & 1) > 0)      {          // case of n = 1 handled separately          if (n == 1)              Console.WriteLine( -1);          else         {              b = (n * n - 1) / 2;              c = (n * n + 1) / 2;              Console.WriteLine( "Side b = " + b                             + ", Side c = " + c);          }      }      else     {          // case of n = 2 handled separately          if (n == 2)              Console.WriteLine( -1);          else         {              b = n * n / 4 - 1;              c = n * n / 4 + 1;              Console.WriteLine( "Side b = " + b +                               ", Side c = " + c);          }      }            // Print angles of the triangle     printAngles(n, b, c); }    // Driver Code public static void Main ()  {     int a = 12;           printOtherSides(a); } }   // This code is contributed  // by inder_verma

## PHP

 

## Javascript

 

Output:

Side b = 35, Side c = 37
Angles are A = 18.9246, B = 71.0754, C = 90

Time Complexity: O(1), since there is no loop or recursion.
Auxiliary Space: O(1), since no extra space has been taken.

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