Question

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

Explanation:

Step1: Identify the triangle type

This is a right - isosceles triangle (two legs are equal). Let the length of each leg be \(a = 11\).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is \(x^{2}=a^{2}+a^{2}\). Substitute \(a = 11\) into the formula:

$$
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Step3: Simplify the radical

Factor \(242\): \(242=2\times11^{2}\). Then \(x=\sqrt{2\times11^{2}}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}(a\geq0,b\geq0)\), we get \(x = 11\sqrt{2}\).

Answer:

\(11\sqrt{2}\)