To find the length of side $x$ in simplest radical form for this square, let's analyze the information provided.

Problem Analysis:

1. The square has a diagonal labeled $\sqrt{8}$.
2. Since the figure is a square, the diagonal splits the square into two 45-45-90 right triangles.
3. In a 45-45-90 triangle, the relationship between the legs (which are equal) and the hypotenuse (diagonal) is as follows: $\text{Hypotenuse} = \text{leg} \times \sqrt{2}$ So, if we let the length of each leg (which corresponds to $x$) be $x$, then we have: $x\sqrt{2} = \sqrt{8}$

Solving for $x$:

1. Divide both sides by $\sqrt{2}$ to isolate $x$: $x = \frac{\sqrt{8}}{\sqrt{2}}$
2. Simplify the right side: $x = \sqrt{\frac{8}{2}} = \sqrt{4} = 2$

Answer:

The length of side $x$ in simplest radical form is: $x = 2$

Would you like further explanation, or do you have any questions?

Here are some related questions to expand understanding:

1. How would you find the side length if the diagonal was given as $\sqrt{18}$ instead?
2. What is the relationship between side lengths and diagonals in a square?
3. How can you determine the area of a square given only the diagonal?
4. What are other properties of 45-45-90 triangles that can be useful in geometry problems?
5. How do you rationalize denominators when simplifying radical expressions?

Tip: In a 45-45-90 triangle, the length of each leg is the hypotenuse divided by $\sqrt{2}$, which is helpful for quickly solving problems involving squares and their diagonals.