Question

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

Explanation:

Step1: Apply Pythagorean theorem

In a square, the diagonal divides it into two right - angled isosceles triangles. Let the side of the square be \(x\). By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = b=x\) and \(c = 4\). So \(x^{2}+x^{2}=4^{2}\).

Step2: Simplify the equation

Combine like terms: \(2x^{2}=16\). Then \(x^{2}=\frac{16}{2}=8\).

Step3: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{8}\). Rationalize and simplify: \(x = 2\sqrt{2}\) (since \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\), and the denominator is \(1\) which is rational).

Answer:

\(x = 2\sqrt{2}\)