Right Triangle: Given hypotenuse and ratio of legs, find legs We are given a Right triangle where the Hypotenuse = $20$ cm.
The opposite side is $3$ times longer than the bottom side. 
Is it possible to calculate the length of the opposite side? 
(Tried substitution)
$$a^2 + b^2 = 400$$
$$a = 3b$$
$$(3b)^2 + b^2 = 400$$
$b = 10$ = not correct

Suppose the length
of the hypotenuse is $c$
and the other two sides
have lengths $a$ and $b$.
We know that
$c^2 = a^2+b^2$.
If
$\dfrac{a}{b} = r$,
then
$a = br$
so that
$c^2
= a^2+b^2
= (br)^2+b^2
= b^2(r^2+1)
$.
Therefore,
if you know $c$ and $r$,
$b^2
=\dfrac{c^2}{r^2+1}
$
and
$a^2 
= b^2r^2
=\dfrac{c^2r^2}{r^2+1}
$.
In your case,
$c=20$ and $r = 3$,
so
$b^2
=\dfrac{20^2}{10}
=40
$
and
$a^2 
= b^2r^2
=40\cdot 9
=360
$.
