## Task 3 - 130713

The perimeter $u$ of an isosceles triangle is to be $24 \mathrm{~cm}$; one of the sides of this triangle is to be $2 \frac{1}{2}$ times as long as another of its sides.

Investigate whether it is possible to specify the side lengths of a triangle in such a way that these conditions are met! Investigate whether there is exactly one such possibility! If this is the case, determine the corresponding side lengths!
If the length $c$ of the base and the length $a$ of a leg of an isosceles triangle satisfy the conditions of the problem, then $c=\frac{5}{2} a$ or $a=\frac{5}{2} c$. If $c=\frac{5}{2} a$, then $a+a<\frac{5}{2} a-c$, which contradicts the triangle inequality.

If $a=\frac{5}{2} c$, then $\frac{5}{2} c+\frac{5}{2} c+c=24 \mathrm{~cm}$, so $6 c=24 \mathrm{~cm}$, from which we get $c=4 \mathrm{~cm}$ and $a=10 \mathrm{~cm}$.

Therefore, only $a=10 \mathrm{~cm}$ and $c=4 \mathrm{~cm}$ can satisfy the conditions of the problem. They indeed satisfy them, as they meet the triangle inequalities and the other conditions.