We are given a 30-60-90 right triangle. The hypotenuse is 5, and we need to find the length of the side $x$ opposite the 30-degree angle. The answer must be in simplest radical form with a rational denominator.

1. Problem Description

We are given a 30-60-90 right triangle. The hypotenuse is 5, and we need to find the length of the side $x$ opposite the 30-degree angle. The answer must be in simplest radical form with a rational denominator.

2. Solution Steps

In a 30-60-90 triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. Let the side lengths be $a$, $a\sqrt{3}$, and $2a$, where $a$ is the side opposite the 30-degree angle, $a\sqrt{3}$ is the side opposite the 60-degree angle, and $2a$ is the hypotenuse.

In this case, the side opposite the 30-degree angle is $x$, and the hypotenuse is

5. So, we have:

$2a = 5$

$a = \frac{5}{2}$

Since $x$ is opposite the 30-degree angle, $x = a$.

Therefore, $x = \frac{5}{2}$.

However, this does not produce the lengths given in the figure, so instead, the side of length 5 is the side opposite of the 60-degree angle, such that

$a\sqrt{3}=5$

$a=\frac{5}{\sqrt{3}}$

Rationalize the denominator by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$.

$a=\frac{5\sqrt{3}}{3}$

Here, $x$ is the length of the side opposite the 30-degree angle, so $x = a$.

Thus, $x = \frac{5\sqrt{3}}{3}$.

3. Final Answer

$\frac{5\sqrt{3}}{3}$