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, if the side - length of the square is \(a\), and the diagonal is \(x\), and we consider half of the square as a right - triangle with legs of length \(\sqrt{10}\). According to the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\). Here \(a = b=\sqrt{10}\), and \(c=x\). So \(x^{2}=(\sqrt{10})^{2}+(\sqrt{10})^{2}\).

Step2: Simplify the right - hand side

\(x^{2}=10 + 10=20\).

Step3: Solve for \(x\)

Take the square root of both sides: \(x=\sqrt{20}\).

Step4: Simplify the radical

\(x=\sqrt{4\times5}=2\sqrt{5}\).

Answer:

\(2\sqrt{5}\)