The problem asks us to find the missing side length of two similar triangles, where corresponding sides are proportional. Here's the analysis:

Step-by-step Solution:

1. Identify given side lengths:

   • Triangle 1: Sides are $10, 14,$ and $3$.
   • Triangle 2: Sides are $40, 56,$ and $x$ (unknown side).

2. Set up a proportion: Since the triangles are similar, the ratios of corresponding side lengths must be equal: $\frac{\text{Side 1 of Triangle 1}}{\text{Side 1 of Triangle 2}} = \frac{\text{Side 2 of Triangle 1}}{\text{Side 2 of Triangle 2}} = \frac{\text{Side 3 of Triangle 1}}{\text{Side 3 of Triangle 2}}$

3. Choose a known ratio: Using $\frac{14}{56}$: $\frac{14}{56} = \frac{3}{x}$

4. Simplify the ratio: $\frac{14}{56} = \frac{1}{4}$

5. Solve for $x$: $\frac{1}{4} = \frac{3}{x} \implies x = 3 \cdot 4 = 12$

Final Answer:

The missing side length is $\mathbf{12}$.

Would you like additional details or have any questions? Here are related concepts to explore:

1. What are the conditions for triangles to be similar?
2. How to calculate missing sides using proportions?
3. What are other methods to solve problems involving similar figures?
4. Can you apply the same technique to other shapes like rectangles?
5. How is the concept of proportionality used in real-life applications?

Tip: Always confirm corresponding sides are proportional before solving problems involving similar triangles.