Two sides of a triangle are 2 and `sqrt3` and the included angle is `30^@` then the in-radius r of the triangle is equal to

Options

A. `1/4 (sqrt3-1)`
B. `1/2(sqrt3+1)`
C. `1/2 (sqrt3-1)`
D. `1/4(sqrt3+1)`

Answer

To find the in-radius \( r \) of the triangle with sides \( A = 2 \), \( B = \sqrt{3} \), and included angle \( C = 30^\circ \), we can follow these steps:

### Step 1: Calculate the Area of the Triangle
The area \( \Delta \) of the triangle can be calculated using the formula:
\[ \Delta = \frac{1}{2} A B \sin C \]
Substituting the values:
\[ \Delta = \frac{1}{2} \times 2 \times \sqrt{3} \times \sin(30^\circ) \]
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[ \Delta = \frac{1}{2} \times 2 \times \sqrt{3} \times \frac{1}{2} = \frac{\sqrt{3}}{2} \]

### Step 2: Find the Third Side \( C \)
To find the length of the third side \( C \), we can use the cosine rule:
\[ \cos C = \frac{A^2 + B^2 - C^2}{2AB} \]
Substituting \( C = 30^\circ \):
\[ \cos(30^\circ) = \frac{A^2 + B^2 - C^2}{2AB} \]
This gives us:
\[ \frac{\sqrt{3}}{2} = \frac{2^2 + (\sqrt{3})^2 - C^2}{2 \times 2 \times \sqrt{3}} \]
Calculating \( A^2 + B^2 \):
\[ \frac{\sqrt{3}}{2} = \frac{4 + 3 - C^2}{4\sqrt{3}} \]
Cross-multiplying:
\[ \sqrt{3} \cdot 4\sqrt{3} = 2(7 - C^2) \]
This simplifies to:
\[ 12 = 14 - 2C^2 \]
Rearranging gives:
\[ 2C^2 = 2 \implies C^2 = 1 \implies C = 1 \]

### Step 3: Calculate the Semi-perimeter \( S \)
The semi-perimeter \( S \) is given by:
\[ S = \frac{A + B + C}{2} = \frac{2 + \sqrt{3} + 1}{2} = \frac{3 + \sqrt{3}}{2} \]

### Step 4: Calculate the In-radius \( r \)
The in-radius \( r \) is given by the formula:
\[ r = \frac{\Delta}{S} \]
Substituting the values we found:
\[ r = \frac{\frac{\sqrt{3}}{2}}{\frac{3 + \sqrt{3}}{2}} = \frac{\sqrt{3}}{3 + \sqrt{3}} \]
To rationalize the denominator:
\[ r = \frac{\sqrt{3}(3 - \sqrt{3})}{(3 + \sqrt{3})(3 - \sqrt{3})} = \frac{3\sqrt{3} - 3}{9 - 3} = \frac{3\sqrt{3} - 3}{6} \]
This simplifies to:
\[ r = \frac{\sqrt{3} - 1}{2} \]

### Final Result
Thus, the in-radius \( r \) of the triangle is:
\[ r = \frac{\sqrt{3} - 1}{2} \]