Chapter 9.8, Problem 6P

To determine

To calculate:

The length of legs of right triangle

Answer to Problem 6P

Length of two legs of the right angel triangle is $7m$ and 24m

Explanation of Solution

Given information:

  $\begin{array}{l}P=56m\\ C=25m\end{array}$

Calculation:

For the right angle triangle:

Use the hypotenuse formula as:

  $\begin{array}{l}{C}^2=x^2+y^2\\ {25}^2=x^2+y^2\end{array}$

  $x^2+y^2={25}^2$  ... (1)

Now, the perimeter of the right angle is given as:

  $x+y+25=56$

  $y=31-x$  ... (2)

Now, put value $y=31-x$ in eq. (1):

  $\begin{array}{l}{x}^2+y^2={25}^2\\ x^2+{\left(31-x\right)}^2={25}^2\\ x^2+961-62x+x^2=625\\ 2x^2-62x+336=0\\ x^2-31x+168=0\end{array}$

Use the roots of quadratic formula to find the roots $x$ :

  $\begin{array}{l}x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-\left(-31\right)\pm \sqrt{{\left(-31\right)}^2-4\left(1\right)\left(168\right)}}{2\left(1\right)}\\ x=\frac{31\pm \sqrt{289}}{2}\\ x=\frac{31\pm 17}{2}\\ x=7or24\end{array}$

Therefore, the length of two legs of the right angel triangle is $7m$ and 24m