Question

The sides of a right-angled triangle containing the right angle are $5x$ cm and $3x-1$ cm. If its area is $60c{m}^2$, find its perimeter.

Answer 1

Area of a right angled triangle

$=\frac{1}{2}\times b\times h$

$\Rightarrow 60=\frac{1}{2}\times \left(5x\right)\times (3x-1)$

$\Rightarrow 24=x(3x-1)$

$\Rightarrow 3x^2-x-24=0$

$\Rightarrow 3x^2-9x+8x-24=0$

$\Rightarrow 3x(x-3)+8(x-3)=0$

$\therefore x=-\frac{8}{3}$, $3$

For $x=3$, AB$=3\left(3\right)-1=9-1=8$

$BC=5\times 3=15$

$A{B}^2+B{C}^2=A{C}^2$

$\Rightarrow A{C}^2=8^2+{15}^2=64+225=289$

$\therefore AC=17$ units(cm)

$\therefore$ perimeter $=AB+BC+CA$

$=8+15+17=40$cm.