The figure in the image shows a square, and we are tasked with finding the length of side $x$ in simplest radical form with a rational denominator.

Given that the square is rotated, it is likely that the line segment labeled 4 represents half the diagonal of the square. The diagonal of a square can be found using the Pythagorean theorem, and the relationship between the side length $s$ and the diagonal $d$ is:

$d = s\sqrt{2}$

Since the diagonal is split into two equal parts, each part would be $\frac{d}{2}$. The length of each part (4 in this case) can be used to find the side length $s$.

Let’s solve this step-by-step:

1. From the diagram, the length of each half of the diagonal is 4. This implies that the full diagonal is:

$d = 4 \times 2 = 8$

2. Using the relationship $d = s\sqrt{2}$, we can substitute the value of the diagonal:

$8 = s\sqrt{2}$

3. Solving for $s$, divide both sides by $\sqrt{2}$:

$s = \frac{8}{\sqrt{2}}$

4. To rationalize the denominator, multiply both the numerator and the denominator by $\sqrt{2}$:

$s = \frac{8}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{8\sqrt{2}}{2} = 4\sqrt{2}$

Thus, the length of side $x$ is $4\sqrt{2}$.

Would you like a more detailed explanation or clarification? Here are some related questions for further exploration:

1. How do you find the diagonal of a square from its side length?
2. What is the process of rationalizing a denominator in a fraction?
3. Can you explain the Pythagorean theorem and its application to squares?
4. How would you calculate the diagonal of a square if the side length were given directly?
5. What is the significance of the term "simplest radical form" in mathematics?

Tip: When simplifying expressions with radicals, always try to rationalize the denominator to avoid irrational numbers in the denominator.