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 (since two angles are \(45^{\circ}\) and one is \(90^{\circ}\)), so the legs are equal and the hypotenuse \(c\) and leg \(a\) are related by \(c = a\sqrt{2}\), or \(a=\frac{c}{\sqrt{2}}\). Here, the hypotenuse is 9, and \(x\) is a leg.

Step2: Apply the 45 - 45 - 90 triangle ratio

For a 45 - 45 - 90 triangle, if the hypotenuse is \(h\) and the leg is \(l\), then \(l=\frac{h}{\sqrt{2}}\). Substituting \(h = 9\), we get \(l=\frac{9}{\sqrt{2}}\).

Step3: Rationalize the denominator

To rationalize \(\frac{9}{\sqrt{2}}\), we multiply the numerator and denominator by \(\sqrt{2}\): \(\frac{9\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{9\sqrt{2}}{2}\)

Answer:

\(\frac{9\sqrt{2}}{2}\)