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 - 45 - 90 right triangle. In a 45 - 45 - 90 triangle, the legs are equal. Let the legs be \(a = 9\) and \(b = 9\).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem is \(a^{2}+b^{2}=c^{2}\). Substitute \(a = 9\) and \(b = 9\) into the formula:

$$
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$$

Step3: Simplify the radical

We know that \(162=81\times2\). So \(x=\sqrt{81\times2}\). Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a = 81\), \(b = 2\)), we get \(x=\sqrt{81}\cdot\sqrt{2}=9\sqrt{2}\).

Answer:

\(9\sqrt{2}\)