In a `DeltaABC`, if `A=120^(@), b=2` and `C=30^(@)`, then a=

Options

A. `2sqrt(3)`
B. 2
C. `(sqrt(3))/(2)`
D. `(1)/(2)`

Explanation

To find the length of side \( a \) in triangle \( ABC \) given \( A = 120^\circ \), \( b = 2 \), and \( C = 30^\circ \), we can use the Law of Sines. Here are the steps to solve the problem:

### Step 1: Find angle \( B \)
Using the triangle angle sum property, we know that the sum of the angles in a triangle is \( 180^\circ \). Therefore, we can find angle \( B \) as follows:

\[ B = 180^\circ - A - C \]
Substituting the values of \( A \) and \( C \):

\[ B = 180^\circ - 120^\circ - 30^\circ = 30^\circ \]

### Step 2: Apply the Law of Sines
The Law of Sines states that:

\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

We will use the first two parts of the equation to find \( a \):

\[ \frac{a}{\sin A} = \frac{b}{\sin B} \]

Substituting the known values:

\[ \frac{a}{\sin 120^\circ} = \frac{2}{\sin 30^\circ} \]

### Step 3: Calculate \( \sin 120^\circ \) and \( \sin 30^\circ \)
We know:

\[ \sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \]
\[ \sin 30^\circ = \frac{1}{2} \]

### Step 4: Substitute the sine values into the equation
Now substituting the sine values back into the equation:

\[ \frac{a}{\frac{\sqrt{3}}{2}} = \frac{2}{\frac{1}{2}} \]

### Step 5: Simplify the right side
Calculating the right side:

\[ \frac{2}{\frac{1}{2}} = 2 \times 2 = 4 \]

### Step 6: Solve for \( a \)
Now we can solve for \( a \):

\[ \frac{a}{\frac{\sqrt{3}}{2}} = 4 \]

Multiplying both sides by \( \frac{\sqrt{3}}{2} \):

\[ a = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \]

### Final Answer
Thus, the length of side \( a \) is:

\[ a = 2\sqrt{3} \]