The problem asks to calculate the diagonal of a rectangle where one side is $5$ and the other side is $5\sqrt{2}$.

We can use the Pythagorean theorem to calculate the diagonal $x$, since the diagonal of a rectangle forms a right triangle with the two sides.

The formula is:

$x^2 = a^2 + b^2$

Where $a$ and $b$ are the sides of the rectangle. Plugging in the values:

$x^2 = 5^2 + (5\sqrt{2})^2$

$x^2 = 25 + (5^2)(\sqrt{2}^2)$

$x^2 = 25 + 25 \times 2 = 25 + 50 = 75$

Taking the square root of both sides:

$x = \sqrt{75} = 5\sqrt{3}$

Thus, the length of the diagonal $x$ is $5\sqrt{3}$.

Let me know if you want any more details or clarifications. Here are some related questions:

1. What is the Pythagorean theorem and how does it apply to other shapes?
2. How do you simplify square roots like $\sqrt{75}$?
3. Can you calculate the diagonal of a square with sides of length $a$?
4. How would you apply the Pythagorean theorem to 3D shapes?
5. What is the relationship between the diagonal and the area of a rectangle?

Tip: The Pythagorean theorem is applicable whenever you're dealing with right triangles, so it's very useful for solving problems involving diagonals in rectangular shapes!