Question

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

Explanation:

Step1: Use the property of a 30 - 60 - 90 triangle

In a 30 - 60 - 90 triangle, if the side opposite the 30° angle is \(a\), the side opposite the 60° angle is \(a\sqrt{3}\), and the hypotenuse is \(2a\). Let the side opposite the 30° angle be \(y\). Assume the hypotenuse is \(2y\), and the side \(x\) (opposite 60°) is \(y\sqrt{3}\).

Step2: Relate to the given side (assuming the side opposite 30° is known or can be set)

Let’s assume the side opposite 30° is \(1\) (for the ratio purpose). But if we consider the general form, if we know the side opposite 30° is \(k\), then \(x = k\sqrt{3}\). If we assume the side opposite 30° is \(1\) (since the ratio is what matters for the radical form), \(x=\sqrt{3}\) (in the simplest radical form with rational denominator as \(\sqrt{3}=\frac{\sqrt{3}}{1}\)).

Answer:

\(\sqrt{3}\)