Question 15 Marks

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

Answer

Let the smaller side of the right triangle be x cm and the longer side by y cm.
Using Pythagoras theorem, we have
$x^2+y^2=(3 \sqrt{5})^2$
$\Rightarrow x^2+y^2=45 \ldots \text { (i) }$

If the smaller side is tripled and larger side is doubled, then
The smaller side $=3 x cm$
Larger side $=2 ycm$
New hypotenuse $=15 cm$
Then by Pythagoras theorem, we have
$\mid(3 x)^2+(2 y)^2=(15)^2$
$\Rightarrow 9 x^2+4 y^2=225 \ldots$ (ii)
From (i), $y 2=45-x 2$ and putting in (ii) we get
$ 9 x^2+4\left(45-x^2\right)=225$
$\Rightarrow 9 x^2+180-4 x^2=225$
$\Rightarrow 5 x^2=225-180=45$
$\Rightarrow x^2=9$
$\Rightarrow x= \pm 3 . $
But $x = -3$ is not possible as length can't be $- ve.$ Then $x = 3$ cm From (i), we have
$ x^2+y^2=45$
$\Rightarrow 9+y^2=45$
$\Rightarrow y^2=36$
$\Rightarrow y= \pm 6 $
Rejecting $- ve$ sign then $y = 6$
Hence, the length of the smaller side $= 3\ cm$
The length of the longer side $= 6\ cm.$

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