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

Explanation:

Step1: Use Pythagorean theorem

For an isosceles right - triangle, \(a = b=x\) and \(c = 4\). By Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), 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: \(x=\sqrt{8}\). Simplify \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\).

Answer:

\(2\sqrt{2}\)