The longest side of a triangle $ABC$ is 18 cm and the shortest side of triangle similar to the triangle $ABC$, $A'B'C'$ is 8/3 cm. If the area $P'$ of the triangle $A'B'C'$ is equal to 12 cm$^2$, what is the area $P$ of the triangle $ABC$?
I've been trying to solve this one and I can't get to the answer, could it be that this is unsolvable?
Let's reformulate the question to start with triangle $A'B'C'$.
Triangle $A'B'C'$ has area 12 cm^2, and a shortest side of length 8/3 cm. Triangle $ABC$ is similar to $A'B'C'$, and has longest side 18 cm. What is its area?
Looking only at $A'B'C'$, we cannot find the length of its longest side. If you consider the short side as its horizontal base, you can calculate its height from the area. The top vertex of the triangle can be moved any amount horizontally without changing the length of the base or of its area. So the length of its longest side can also vary.
This means that there is no way to determine what the scaling factor is between the triangles to make the longest side of $A'B'C'$ become of length 18 cm. The area of $ABC$ therefore cannot be determined from the given information alone.
If however you also know that the triangles are right triangles, then it can be determined because then the apex of $A'B'C'$ is fixed.