Question

The hypotenuse of a right triangle is $3\sqrt{5}cm$. If the smaller side is tripled and the larger side is doubled, the new hypotenuse will be $15cm$. Find the length of each side.

Answer 1

Let the smaller side of the right triangle be $x$ cm and the larger side by $y$ cm.

Then, $x^2+y^2={\left(3\sqrt{5}\right)}^2$

$\Rightarrow x^2+y^2=\mathrm{45....}\left(i\right)$

If the smaller side is tripled and the larger side be doubled, the new hypotenuse is $15cm$.

$\therefore {\left(3x\right)}^2+{\left(2y\right)}^2={15}^2\Rightarrow 9x^2+4y^2=\mathrm{225...}\left(ii\right)$

From equation (i) we get $y^2=45-x^2$

Putting $y^2=45-x^2$ in equation (ii), we get

$9x^2+4(45-x^2)=225\Rightarrow 5x^2+180=225\Rightarrow 5x^2=45\Rightarrow x^2=9\Rightarrow x=\pm 3$

But length of a side cannot be negative. Therefore $x=3$

Putting $x=3$ in (i) we get

$9+y^2=45\Rightarrow y^2=36\Rightarrow y=6$

Hence, the length of the smaller side is $3cm$ and the length of the larger side is $6cm$.