## Task $7 / 83$

In a right-angled triangle, the side lengths are integer multiples of the unit length. Furthermore, it is given that the perimeter is numerically equal to the area. Determine all such triangles!
If the legs of the sought triangles are denoted by $a$ and $b$, and the hypotenuse by $c$, then according to the problem statement:

1. $a ; b ; c \in \mathbb{N}$
2. $a^{2}+b^{2}=c^{2}$, thus $c=\sqrt{a^2+b^2}$
3. $a+b+c=0.5ab$

Eliminating the hypotenuse $c$ from the equations and solving the resulting Diophantine equation for $a$, we obtain:

$$
a=4+\frac{8}{b-4}
$$

Since $a ; b \in \mathbb{N}$, the only solutions are $b_{1}=12, a_{1}=5, b_{2}=8, a_{2}=6, b_{3}=6, a_{3}=8, b_{4}=5, a_{4}=12$. Due to the symmetry of the equations in $a$ and $b$, the last two solutions can be discarded. Therefore, there are exactly 2 triangles of the desired type:

1. $a_{1}=5 ; b_{1}=12, c_{1}=13$ with $a_{1}+b_{1}+c_{1}=30=0.5 a_{1} b_{1}$
2. $a_{2}=6 ; b_{2}=8, c_{2}=10$ with $a_{2}+b_{2}+c_{2}=24=0.5 a_{2} b_{2}$