Question

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

Explanation:

Step1: Recall the property of a square

In a square, the diagonal \(x\) and the side \(a = \sqrt{8}\) are related by the formula \(x=\sqrt{2}a\) (derived from the Pythagorean theorem \(x^{2}=a^{2}+a^{2}=2a^{2}\)).

Step2: Substitute the value of \(a\)

Substitute \(a = \sqrt{8}\) into the formula \(x=\sqrt{2}a\). So \(x=\sqrt{2}\times\sqrt{8}\).
Using the property \(\sqrt{m}\times\sqrt{n}=\sqrt{mn}\), we have \(x=\sqrt{2\times8}=\sqrt{16}\). But we can also simplify \(\sqrt{8}\) first. Since \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\). Then \(x=\sqrt{2}\times2\sqrt{2}\).
Using the property \(\sqrt{m}\times\sqrt{m}=m\), we get \(x = 2\times(\sqrt{2}\times\sqrt{2})=2\times2 = 4\). Another way: \(x=\sqrt{2}\times\sqrt{8}=\sqrt{2}\times2\sqrt{2}=2\times2=4\)

Answer:

\(4\)