A triangle has two sides, whose lengths differ by $12 \mathrm{~cm}$, and two sides, whose lengths differ by $15 \mathrm{~cm}$. The perimeter of this triangle is $75 \mathrm{~cm}$. Determine the lengths of its sides. Find all possible options.

(L. Šimunek)

Hint. Denote the length of one side as an unknown and then express the lengths of the other sides using it. Consider how many possibilities you need to discuss.
Let's denote the lengths of two sides, which differ by $12 \mathrm{~cm}$, as $s$ and $s+12$. The third side differs from one of these two by $15 \mathrm{~cm}$. It is not specified from which one and whether it is $15 \mathrm{~cm}$ greater or smaller; therefore, we must discuss the following four possibilities:

|  | lengths of the triangle sides |  |  |
| :---: | :---: | :---: | :---: |
| 1. possibility | $s$ | $s+12$ | $s+15$ |
| 2. possibility | $s$ | $s+12$ | $s-15$ |
| 3. possibility | $s$ | $s+12$ | $s+12+15$ |
| 4. possibility | $s$ | $s+12$ | $s+12-15$ |

The perimeter of the triangle is $75 \mathrm{~cm}$. Therefore, for each possibility, we will set up an equation and solve for the corresponding $s$. As an example, we will only show the calculation for the 1st possibility:

$$
\begin{aligned}
s+(s+12)+(s+15) & =75 \\
3 s+27 & =75, \\
3 s & =48, \\
s & =16 .
\end{aligned}
$$

We then substitute the resulting $s$ into the table:

|  | lengths of the triangle sides |  |  |
| :---: | :---: | :---: | :---: |
| 1. possibility | 16 | 28 | 31 |
| 2. possibility | 26 | 38 | 11 |
| 3. possibility | 12 | 24 | 39 |
| 4. possibility | 22 | 34 | 19 |

To ensure that the calculated values actually correspond to the sides of a triangle, the triangle inequality must hold. Therefore, we still need to check whether the largest number in each row is less than the sum of the other two. This inequality holds only for the 1st and 4th possibilities. Thus, the problem has two solutions: the lengths of the triangle sides can be $16 \mathrm{~cm}, 28 \mathrm{~cm}$, and $31 \mathrm{~cm}$, or $19 \mathrm{~cm}, 22 \mathrm{~cm}$, and $34 \mathrm{~cm}$.