Question

question\
the figure below is a square. find the length of side x in simplest radical form with a rational denominator.

Explanation:

Step1: Identify triangle properties

The square's diagonal splits it into two isosceles right triangles, where the diagonal is the hypotenuse, and each leg is length $x$.

Step2: Apply Pythagorean theorem

For a right triangle, $a^2 + b^2 = c^2$. Here, $a = b = x$, $c = 4$.
$$x^2 + x^2 = 4^2$$

Step3: Simplify the equation

Combine like terms and solve for $x^2$.
$$2x^2 = 16$$
$$x^2 = 8$$

Step4: Solve for x and rationalize

Take the square root, then simplify to rational form.
$$x = \sqrt{8} = 2\sqrt{2}$$

Answer:

$2\sqrt{2}$