To solve this problem, we are determining the value of $x$, which represents the diagonal (or hypotenuse) of the square. The square has side lengths of $1$ unit.

The diagonal of a square can be calculated using the Pythagorean theorem:

$x^2 = 1^2 + 1^2$

Simplify the equation:

$x^2 = 1 + 1 = 2$

Now take the square root of both sides:

$x = \sqrt{2}$

Thus, the value of $x$ is:

$x = \sqrt{2} \approx 1.414$

Would you like additional clarification or a breakdown of the steps?

Relative Questions:

1. Why is the diagonal of a square determined using the Pythagorean theorem?
2. How does the diagonal relate to the side length of any square?
3. Can we generalize this diagonal calculation for squares of different side lengths?
4. What is the importance of square roots in geometry, particularly in this problem?
5. How would this calculation change if the shape were a rectangle?

Tip:

Remember, the diagonal of a square always equals the side length multiplied by $\sqrt{2}$.