Question

The perimeter of a right triangle is $60$ cm, its hypotenuse is $25$ cm. Find the area of the triangle.

Answer 1

In right angled $△PQR$, using pythagoras theorem,

$a^2+b^2={25}^2=625⟶\left(1\right)$

and

$a+b+25=60cm$

$\Rightarrow a+b=35$

$\Rightarrow a=35-b$

Put value of $a$ in equation (1), we get

${(35-b)}^2+b^2=625$

$\Rightarrow 1225+b^2-70b+b^2=625$

$\Rightarrow {2b}^2-70b+125-625=0$

$\Rightarrow {2b}^2-70b+600=0$

$\Rightarrow b^2-35b+300=0$

$\Rightarrow b^2-20b-15b+300=0$

$\Rightarrow (b-20)(b-15)=0$

$\Rightarrow$ $b=20or15$

if $b=20$, then $a=35-20=15$

& if $b=15$ then $a=35-15=20$

Therefore, ar $△PQR=\frac{1}{2}\times a\times b=\frac{1}{2}\times 15\times 20=150{cm}^2$.