Problem 3. The perimeter of a triangle is $34 \mathrm{~cm}$. If one of the sides is increased by $3 \mathrm{~cm}$, another is increased by $2 \mathrm{~cm}$, and the third is decreased by $3 \mathrm{~cm}$, a new triangle is obtained which is equilateral. Determine the length of the side of the newly obtained triangle.
Solution. First method. Since the perimeter of a triangle increases or decreases proportionally to the increase or decrease in the length of one of its sides, the perimeter of the new triangle will be equal to $34+3+2-3=36 \mathrm{~cm}$. However, this triangle is equilateral, so the length of its side is equal to $36: 3=12 \mathrm{~cm}$.

Second method. Let the length of the side of the resulting equilateral triangle be $x$. Then, for the lengths of the sides $a, b, c$ of the initial triangle, whose perimeter is $a+b+c=34$, it holds that $a+3=x, b+2=x, c-3=x$. This means that

$$
\begin{aligned}
& 3 x=a+3+b+2+c-3 \\
& 3 x=(a+b+c)+3+2-3 \\
& 3 x=34+3+2-3 \\
& 3 x=36 \\
& x=12 \mathrm{~cm}
\end{aligned}
$$