Question

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

Explanation:

Step1: Identify the triangle type

The triangle has two angles of \(45^\circ\) and one right angle (\(90^\circ\)), so it's a 45 - 45 - 90 triangle. In a 45 - 45 - 90 triangle, the legs are equal, and the hypotenuse \(h\) is related to the leg length \(l\) by \(h = l\sqrt{2}\), or the leg length \(l=\frac{h}{\sqrt{2}}\) (rationalizing the denominator, \(l = \frac{h\sqrt{2}}{2}\)). Here, the hypotenuse is 1, and we want to find the length of leg \(x\).

Step2: Apply the 45 - 45 - 90 triangle ratio

For a 45 - 45 - 90 triangle, if the hypotenuse is \(h\) and the leg is \(x\), then \(x=\frac{h}{\sqrt{2}}\). Given \(h = 1\), we have \(x=\frac{1}{\sqrt{2}}\). To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{2}\): \(x=\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{2}}{2}\).

Answer:

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