Question

find the length of side ( x ) in simplest radical form with a rational denominator.

Explanation:

Step1: Identify the triangle type

The triangle is a right - isosceles triangle (two legs are equal as indicated by the tick marks) with hypotenuse \(c = 4\) and legs \(a=x\) and \(b = x\). We use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\).
Substitute \(a = x\), \(b=x\) and \(c = 4\) into the formula: \(x^{2}+x^{2}=4^{2}\)

Step2: Simplify the equation

Combine like terms: \(2x^{2}=16\)
Divide both sides by 2: \(x^{2}=\frac{16}{2}=8\)

Step3: Solve for x

Take the square root of both sides: \(x=\sqrt{8}\)
Simplify \(\sqrt{8}\) by factoring out the perfect square: \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\) (since \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) for \(a = 4\), \(b = 2\) and \(\sqrt{4}=2\))

Answer:

\(2\sqrt{2}\)