\section*{Exercise 3 - 221023}

For a right-angled triangle, the following is required:

(1) The perimeter of the triangle is \(132 \mathrm{~cm}\).

(2) The sum of the areas of the squares over the three sides of the triangle is \(6050 \mathrm{~cm}^{2}\). Prove that there are right-angled triangles that satisfy requirements (1) and (2), and that the lengths of the sides of the triangle are uniquely determined by these requirements! Provide these side lengths!
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A triangle with side lengths \(a, b, c\) measured in centimeters is a right triangle with \(c\) as the length of the hypotenuse if and only if \(a^{2}+b^{2}=c^{2} \quad(3)\) holds, according to the Pythagorean theorem and its converse.

It satisfies (1) and (2) if and only if, in addition, the equations

\[
\begin{aligned}
a+b+c & =132 \\
a^{2}+b^{2}+c^{2} & =6050
\end{aligned}
\]

hold.

I. If (3), (4), and (5) are satisfied, then:

From (3) and (5), it follows that \(2 c^{2}=6050, c^{2}=3025\), and since \(c>0\), \(c=55\). Thus, from (4) and (3),

\[
\begin{aligned}
a+b & =77 \\
a^{2}+b^{2} & =3025
\end{aligned}
\]

From (6), it follows that \(b=77-a\) and thus from (7) \(a^{2}-77 a+1452=0 \quad\) (8) with the solution \(a=\frac{77}{2} \pm \frac{11}{2}\), i.e., either \(a=44\) and \(b=33\) or \(a=33\) and \(b=44\).

Therefore, only the lengths of the legs \(33 \mathrm{~cm}, 44 \mathrm{~cm}\) and the hypotenuse \(55 \mathrm{~cm}\) can satisfy the conditions (1) and (2).

A check shows that they satisfy the conditions. Thus, the required proof is complete.