The figures in the image are similar, meaning the corresponding sides of the figures have the same ratio. To find the missing side $x$, we can use the concept of similarity.

Step-by-step process:

1. Identify corresponding sides:

   • The side of length 7 in the smaller figure corresponds to the side of length 91 in the larger figure.
   • The side of length 4 in the smaller figure corresponds to the side of length 52 in the larger figure.

2. Set up the proportion: Since the figures are similar, the ratio of the corresponding sides should be equal. So we can set up the proportion using the given sides: $\frac{7}{91} = \frac{4}{52} = \frac{11}{x}$

3. Solve for $x$: Using the ratio of the corresponding sides, we first simplify: $\frac{7}{91} = \frac{1}{13} \quad \text{and} \quad \frac{4}{52} = \frac{1}{13}$ So, the scale factor between the smaller and larger figure is $\frac{1}{13}$.

   Now, apply this ratio to find $x$: $\frac{11}{x} = \frac{1}{13}$

4. Cross multiply to solve for $x$: $11 \times 13 = x$ $x = 143$

Final Answer:

The value of the missing side $x$ is 143.

Would you like further details or explanations on any part of the solution?

Here are five related questions you could explore:

1. What is the difference between similarity and congruence in geometry?
2. How do you find the scale factor between two similar polygons?
3. Can triangles also be similar, and how are their properties different from quadrilaterals?
4. What role do angles play in determining if two figures are similar?
5. How can you verify if a set of given figures are similar using only side lengths?

Tip: When working with similar figures, always ensure the corresponding sides are correctly matched before solving the proportion.