Dado un lado del triángulo de ángulo recto, verifique si existe un triángulo de ángulo recto posible con otros dos lados del triángulo. Si es posible, imprima la longitud de los otros dos lados y todos los ángulos del triángulo.
Ejemplos:
Entrada : a = 12
Salida : Los lados son a = 12, b = 35, c = 37
Los ángulos son A = 18,9246, B = 71,0754, C = 90
Explicación : a = 12, b = 35 y c = 37 forman un
triángulo rectángulo porque
12*12 + 35*35 = 37*37
Entrada : a = 6
Salida : Los lados son a = 6, b = 8, c = 10
Los ángulos son A = 36.8699, B = 53.1301, C = 90
Enfoque para verificar si existe un triángulo y encontrar lados :
para resolver este problema, primero observamos la ecuación de Pitágoras. Si a y b son las longitudes de los catetos de un triángulo rectángulo y c es la longitud de la hipotenusa, entonces la suma de los cuadrados de las longitudes de los catetos es igual al cuadrado de la longitud de la hipotenusa.
Esta relación está representada por la fórmula:
a*a + b*b = c*c
Caso 1: a es un número impar: Dado a, encuentra b y c
c2 - b2 = a2 OR c = (a2 + 1)/2; b = (a2 - 1)/2;
La solución anterior funciona solo para el caso en que a es impar, porque a2 + 1 es divisible por 2 solo para impar a.
Caso 2: a es un número par: cuando cb es 2 y c+b es (a2)/2
c-b = 2 & c+b = (a2)/2 Hence, c = (a2)/4 + 1; b = (a2)/4 - 1;
Esto funciona cuando a es par.
Enfoque para encontrar ángulos :
primero encuentre todos los lados del triángulo. Luego se aplicó la regla «SSS» que significa ley del coseno:
![\[1.\ cos(a)=\frac{b^2+c^2-a^2}{2bc}\]\ \[2.\ cos(b)=\frac{a^2+c^2-b^2}{2ac}\]\ \[3.\ cos(c)=\frac{a^2+b^2-c^2}{2ab}\]](https://www.geeksforgeeks.org/wp-content/ql-cache/quicklatex.com-222b3af52b1c9cb78e7607d0cc226415_l3.png "Rendered by QuickLaTeX.com")
A continuación se muestra la implementación del enfoque anterior:
C++
// C++ program to print all sides and angles of right
// angle triangle given one side
#include <bits/stdc++.h>
#include <cmath>
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
<?php
// PHP program to print all sides
// and angles of right angle triangle
// given one side
$PI = 3.1415926535;
// Function to find angle A
// Angle in front of side a
function findAnglesA($a, $b, $c)
{
global $PI;
// applied cosine rule
$A = acos(($b * $b + $c *
$c - $a * $a) / (2 * $b * $c));
// convert into degrees
return $A * 180 / $PI;
}
// Function to find angle B
// Angle in front of side b
function findAnglesB($a, $b, $c)
{
global $PI;
// applied cosine rule
$B = acos(($a * $a + $c *
$c - $b * $b) / (2 * $a * $c));
// convert into degrees and return
return $B * 180 / $PI;
}
// Function to print all angles
// of the right angled triangle
function printAngles($a, $b, $c)
{
$x = (double)$a;
$y = (double)$b;
$z = (double)$c;
// for calculate angle A
$A = findAnglesA($x, $y, $z);
// for calculate angle B
$B = findAnglesB($x, $y, $z);
echo "Angles are A = " . $A .
", B = " . $B . ", C = 90\n";
}
// Function to find other two sides
// of the right angled triangle
function printOtherSides($n)
{
// if n is odd
if ($n & 1)
{
// case of n = 1 handled separately
if ($n == 1)
echo "-1\n";
else
{
$b = ($n * $n - 1) / 2;
$c = ($n * $n + 1) / 2;
echo "Side b = " . $b .
", Side c = " . $c . "\n";
}
}
else
{
// case of n = 2 handled separately
if ($n == 2)
echo "-1\n";
else
{
$b = $n * $n / 4 - 1;
$c = $n * $n / 4 + 1;
echo "Side b = " . $b .
", Side c = " . $c . "\n";
}
}
// Print angles of the triangle
printAngles($n, $b, $c);
}
// Driver Code
$a = 12;
printOtherSides($a);
// This code is contributed by mits
?>
Javascript
<script>
// Javascript program to print all sides and angles of right
// angle triangle given one side
let PI = 3.1415926535;
// Function to find angle A
// Angle in front of side a
function findAnglesA(a, b, c)
{
// applied cosine rule
let 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
function findAnglesB(a, b, c)
{
// applied cosine rule
let 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
function printAngles(a, b, c)
{
let x = a;
let y = b;
let z = c;
// for calculate angle A
let A = findAnglesA(x, y, z);
// for calculate angle B
let B = findAnglesB(x, y, z);
document.write( "Angles are A = " + A + ", B = " +
B + ", C = " + 90);
}
// Function to find other two sides of the
// right angled triangle
function printOtherSides(n)
{
let b=0,c=0;
// if n is odd
if ((n & 1)>0)
{
// case of n = 1 handled separately
if (n == 1)
document.write( -1);
else
{
b = (n*n-1)/2;
c = (n*n+1)/2;
document.write( "Side b = " + b
+ ", Side c = " + c );
}
}
else
{
// case of n = 2 handled separately
if (n == 2)
document.write( -1);
else
{
b = n*n/4-1;
c = n*n/4+1;
document.write( "Side b = " + b
+ ", Side c = " + c + "<br/>");
}
}
// Print angles of the triangle
printAngles(n,b,c);
}
// Driver Code
let a = 12;
printOtherSides(a);
</script>
Producción:
Side b = 35, Side c = 37 Angles are A = 18.9246, B = 71.0754, C = 90
Complejidad de Tiempo : O(1)
Espacio Auxiliar : O(1)