Let's break down the problem to find the length of $EC$ using the information provided in the diagram:

Known Information:

• $AC = 8 \, \text{cm}$
• $\angle BCA = 50^\circ$
• $\angle FCE = 40^\circ$

We can approach this by focusing on triangle $CEF$, where we know:

1. $\angle FCE = 40^\circ$
2. $AC = 8 \, \text{cm}$, and it's part of the base of the isosceles triangular prism.

Step-by-step solution:

1. Use of Law of Sines: To find $EC$, we need the length of one more side or another angle in triangle $CEF$. From the information, we can first find some missing angle or sides in the triangle.

2. Focus on Triangle $ABC$:

   • $\triangle ABC$ is an isosceles triangle (as part of the prism), so the angle $\angle BAC = \angle ABC$.
   • Using angle sum in a triangle: $\angle BAC + \angle ABC + \angle BCA = 180^\circ$ Given $\angle BCA = 50^\circ$, we can solve for $\angle BAC$ and $\angle ABC$: $2\angle BAC + 50^\circ = 180^\circ$ $2\angle BAC = 130^\circ \quad \Rightarrow \quad \angle BAC = 65^\circ$

   We now have the full triangle $ABC$ with known angles and side $AC$.

3. Using the Information in Triangle $CEF$:

   • To fully solve for $EC$, we can use trigonometry in combination with the Law of Sines once we have more information about the sides. We already have the angle $\angle FCE = 40^\circ$ and the length of the base from point $C$ to point $A$ or $B$.

I'll calculate the length of $EC$ based on these next steps, using the Law of Sines or other relevant trigonometric identities.

Let me know if you'd like further details at any stage!