Questions: Find the length of side x in simplest radical form with a rational denominator.

Transcript text: Find the length of side $x$ in simplest radical form with a rational denominator.

Solution

Solution Steps

Step 1: Identify the given information

We are given a right triangle with angles $30^\circ$ and $60^\circ$. The side opposite the $30^\circ$ angle has length 2. We are asked to find the length of the hypotenuse, which is denoted by $x$.

Step 2: Use the 30-60-90 triangle rule

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

Step 3: Set up a proportion

Since the side opposite the $30^\circ$ angle is 2, we can set up a proportion to find the length of the hypotenuse, $x$: $\frac{1}{2} = \frac{2}{x}$

Step 4: Solve for x

Cross-multiply to solve for $x$: $x = 2 \times 2$ $x = 4$