Question

The length of the sides forming a right angle of a right-angled triangle is $5x$ cm and $(3x-1)$ cm. If the area of the triangle is $60{cm}^2$, find its hypotenuse.

Answer 1

Let $ABC$ be a right angled triangle with right angle at $B$

let $AB=5x$ and $BC=3x-1$.

Then, area $=△ABC=\frac{1}{2}(Base\times Height))$

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

$\Rightarrow 120=5x(3x-1)\Rightarrow 3x^2-x-24=0\Rightarrow 3x^2-9x+8x-24=0$

$\Rightarrow (x-3)(3x+8)=0\Rightarrow x=3or3x+8=0\Rightarrow x=3orx=-\frac{8}{3}\Rightarrow x=3$

$AB=5x=5\times 3=15cm$ and $BC=(3x-1)=(3\times 3-1)=8cm$

Now, ${AC}^2+{AB}^2+{BC}^2\Rightarrow {AC}^2={15}^2+8^2\Rightarrow {AC}^2=289\Rightarrow AC=17cm$

Hence, hypotenuse $=17cm$