Item, let's consider a right-angled triangle that satisfies the following conditions:
1. The length of one of the legs of $\triangle$ is an integer 3
2. If the perimeter of $\triangle$ is $x$ cm, then the area of $\Delta$ is $x$ square cm. Determine the lengths of the three sides of $\triangle$.
Let $\triangle$ have the lengths of its two legs as $a, b$, then
$$
\begin{array}{l}
\frac{1}{2} a b=a+b+\sqrt{a^{2}+b^{2}}, \\
4\left(a^{2}+b^{2}\right)=(2 a+2 b-a b)^{2}, \\
a^{2} b^{2}-4 a^{2} b-4 a b^{2}+8 a b=0, \\
a b-4 a-4 b+8=0, \\
(a-4)(b-4)=8 .
\end{array}
$$
Since $a, b$ are natural numbers, therefore
(1) when $a-4=1$, $b-4=8$;
(2) when $a-4=2$, $b-4=4$;
(3) when $a-4=4$, $b-4=2$,
(4) when $a-4=8$, $b-4=1$.
The possible pairs of $(a, b)$ are $(5,12),(6,8),(8,6),(12,5)$. Therefore, the three sides of $\triangle$ are $5,12,13$ or $6,8,10$.