Question

The hypotenuse of a right-triangle is $3\sqrt{10}\text{cm}$. If the smaller side is tripled and the longer side is doubled, the new hypotenuse will be $9\sqrt{5}\text{cm}$. Find the length of each side.

Answer 1

Length of hypotensue$=3\sqrt{10}$

Let the length of shorter side = $x$

and length of longer side = $y$

According to pythagoras theorem,

$x^2+y^2={\left(3\sqrt{10}\right)}^2=90$

$x^2+y^2=90......\left(1\right)$

The smaller side is tripled $=3x$

The larger side is doubled $=2y$

New hypotenuse$=9\sqrt{5}$

Now,

${\left(3x\right)}^2+{\left(2y\right)}^2={\left(9\sqrt{5}\right)}^2$

$9x^2+4y^2=405$

$5x^2+4x^2+4y^2=405$

$5x^2+4(x^2+y^2)=405$

$5x^2+4\times 90=405$

$5x^2=405-360$

$x^2=\frac{45}{5}$

$x^2=9$

$x=3$

Put $x=3$ in (1) and we get,

$y=9$

Hence, this is the answer.