Question

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

Explanation:

Step1: Identify the triangle type

This is a \(45^{\circ}-45^{\circ}-90^{\circ}\) right - triangle. In a \(45^{\circ}-45^{\circ}-90^{\circ}\) triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Let the legs be of length \(a\) and the hypotenuse be \(c\). The formula is \(c = a\sqrt{2}\).

Step2: Apply the formula

Here, one of the legs (let's assume the leg adjacent to the \(45^{\circ}\) angle) can be found using the cosine function. \(\cos45^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}\). We know \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), and the hypotenuse is \(9\). Let the adjacent side (which is equal to the other leg in a \(45 - 45-90\) triangle) be \(x\). So \(\frac{\sqrt{2}}{2}=\frac{x}{9}\).
Cross - multiply: \(x = 9\times\frac{\sqrt{2}}{2}\). But if we use the side - ratio formula (since in a \(45 - 45-90\) triangle, if the hypotenuse \(c = 9\) and \(c=a\sqrt{2}\), then \(a=\frac{c}{\sqrt{2}}\)). Rationalize the denominator: \(x=\frac{9}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{9\sqrt{2}}{2}\)

Answer:

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