Task 3 - 330713 Anke reports that she has drawn an isosceles triangle with a perimeter of $14 \mathrm{~cm}$, in which one of the three sides is exactly three times as long as a second of the three sides.

Beate thinks that the lengths of all three sides are uniquely determined by these specifications.

Christin, on the other hand, believes that the specifications could apply to more than one possibility for the three side lengths.

Investigate whether Beate or Christin is right! Determine all possible options for the three lengths!
If $c$ is the length of the base and $a=b$ is the length of the legs of the mentioned triangle, then according to Anja's statements, either $a=3 \cdot c$ or $c=3 \cdot a$ holds.

Since the triangle inequality $a+a>c$ must hold, the case $c=3 \cdot a$ is excluded; thus, only the possibility $a=3 \cdot c$ remains. (1) According to the given information, $2 \cdot a+c=14 \mathrm{~cm}$ also holds. (2)

From (1) and (2), it follows that $6 \cdot c+c=14 \mathrm{~cm}$, so $c=2 \mathrm{~cm}$ and thus $a=b=6 \mathrm{~cm}$.

These lengths not only satisfy the requirement $a+a>c$, but also the requirement $a+c>a$ of the triangle inequality. Therefore, Beate is right; there is exactly the mentioned possibility for the lengths.