So here's the question :

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

$\sqrt{a^2+b^2}=3\sqrt{5}$ and $\sqrt{(3a)^2+(2b)^2}=15$. A problem with the way you phrase your question is that it looks as if you're passing on to us a question that someone else wrote rather than asking your own question, so it's as if you're doing stenography. – Michael Hardy Dec 20 '11 at 18:16
For the smaller triangle you have $$\sqrt{a^2+b^2} = 3\sqrt{5},$$ so $$a^2+b^2 = 3^2\cdot 5 = 45.\tag{1}$$ For the larger triangle you need $$\sqrt{(3a)^2+(2b)^2} = 15,$$ so $$9a^2+4b^2 = 15^2 = 225.\tag{2}$$ If you say $a^2=45-b^2$, and consequently put $45-b^2$ in place of $a^2$ in (2), then it's easy to solve that for $b^2$, and then put that solution for $b^2$ into $45-b^2$ to find $a^2$.