Question

The perimeter of a right angled triangle is $30cm$ and its hypotenuse is $13cm$. Find the length of the other two sides.

Answer 1

Given :

Perimeter of a right angled triangle $=30cms$.

We know, Perimeter of a right angled triangle $=a+b+c$ and $a^2+b^2=c^2$

where $a,b$ are two sides and $c$ is the hypotenuse of a triangle.

Also, it is given $c=13cms$.

$\therefore$ $a+b+13=30$ and $a^2+b^2={13}^2$

$⟹$ $a+b=17$ ........ $\left(1\right)$ and $a^2+b^2=169$ ............ $\left(2\right)$

From $\left(1\right)$, $a=17-b$

Substitute value of $a$ in equation $\left(2\right)$ we get,

$(17-b{)}^2+b^2=169$

$⟹$ $289-34b+b^2+b^2=169$

$⟹$ $2b^2-34b+120=0$

$⟹$ $b^2-17b+60=0$

$⟹$ $b^2-12b-5b+60=0$ ...... factorized above equation

$⟹$ $b(b-12)-5(b-12)=0$

$⟹$ $(b-12)(b-5)=0$

$\therefore$ $b=12orb=5$

If $b=12$ then $a=17-12=5$

If $b=5$ then $a=17-5=12$.

Hence the length of other two sides are $12,5$.