Question

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

Explanation:

Step1: Identify the triangle type

This is a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle (isosceles right - triangle). The ratio of sides in a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle is \(x:x:x\sqrt{2}\), where the hypotenuse is \(x\sqrt{2}\) and the legs are \(x\).

Step2: Set up the equation

Given the hypotenuse \(c = 4\), and \(c=x\sqrt{2}\). So, \(x\sqrt{2}=4\).

Step3: Solve for \(x\)

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Answer:

\(2\sqrt{2}\)