A triangle has two sides of length 12 and 18. Another triangle, similar but not congruent to it, also has two sides of length 12 and 18. Determine the lengths of the third sides of the triangles.
Solution. The triangles are certainly not isosceles, because in that case they would also be congruent. Let's assume that $a<b<c$ and take two corresponding triangles. Since the two triangles are not congruent, the corresponding sides in the similarity are not of equal length. On the other hand, we have to place two numbers in three positions, so there will be a side whose length is known in both triangles. This can only be the two values of the $b$ side. From this, it is immediately clear that the problem has only one solution. In one triangle, $a=12, b=18$, while in the other, $b=12$ and $c=18$. The similarity ratio $\lambda=\frac{18}{12}=\frac{3}{2}$. From the similarity ratio, the sides of the triangles are 12, 18, 27, and 8, 12, 18.