The image contains several math problems. I will solve problem number 12. The problem asks to find the length of side $x$ in the given triangle in simplest radical form with a rational denominator. The triangle is a right triangle with a 30-60-90 angles, and one side with length 4.

1. Problem Description

The image contains several math problems. I will solve problem number

1

2. The problem asks to find the length of side $x$ in the given triangle in simplest radical form with a rational denominator. The triangle is a right triangle with a 30-60-90 angles, and one side with length

4.

2. Solution Steps

Let's denote the sides of the triangle as follows:

- Hypotenuse: $c = 4$

- Side opposite to 30-degree angle: $x$

- Side opposite to 60-degree angle: unknown

Since it's a 30-60-90 triangle, we know the ratios of the sides are $1:\sqrt{3}:2$.

The hypotenuse is $2x$, where $x$ is the side opposite the 30-degree angle. Therefore, $2x=4$, implying $x=2$.

However, this is not consistent with the problem where $x$ is supposed to be adjacent to 60 degrees. Let's consider the other relationship.

The hypotenuse is

4. The side adjacent to the 60-degree angle which is $x$.

$\cos(60^\circ) = \frac{x}{4}$

Since $\cos(60^\circ) = \frac{1}{2}$, we have $\frac{1}{2} = \frac{x}{4}$.

Solving for $x$, we get $x = 2$.

Let's consider that x is the adjacent side to 30 degrees.

$\cos(30^\circ) = \frac{x}{4}$

Since $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2} = \frac{x}{4}$.

Solving for $x$, we get $x = 2\sqrt{3}$. Since the problem shows $x$ is opposite the 30-degree angle in a right triangle with hypotenuse 4, the correct solution is

$\sin(30^\circ) = \frac{x}{4}$.

Since $\sin(30^\circ) = \frac{1}{2}$, we have $\frac{1}{2} = \frac{x}{4}$.

Solving for $x$, we get $x = 2$.

However, the figure shown in question 12 does not show that. Thus let's assume the 60 degree angle.

$\sin(60^\circ) = \frac{opp}{hyp} = \frac{opp}{4}$

$opp = 4\sin(60^\circ) = 4 * \frac{\sqrt{3}}{2} = 2\sqrt{3}$. Since it shows a 30 60 90, this looks like $x$ is opposite the 30 degree angle.

3. Final Answer

$2\sqrt{3}$