The radius of the incircle of an equilateral `Delta ABC` of side `2sqrt(3)` units is x cm. The value of x is :

Options

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

Answer

To find the radius of the incircle (inradius) of an equilateral triangle with a given side length, we can follow these steps:

### Step 1: Identify the side length of the triangle.
The side length of the equilateral triangle \( ABC \) is given as \( 2\sqrt{3} \) units.

### Step 2: Calculate the semi-perimeter of the triangle.
The semi-perimeter \( s \) of an equilateral triangle is calculated using the formula:
\[ s = \frac{3a}{2} \]
where \( a \) is the length of one side. Thus,
\[ s = \frac{3 \times 2\sqrt{3}}{2} = 3\sqrt{3} \text{ units.} \]

### Step 3: Calculate the area of the triangle.
The area \( A \) of an equilateral triangle can be calculated using the formula:
\[ A = \frac{\sqrt{3}}{4} a^2 \]
Substituting \( a = 2\sqrt{3} \):
\[ A = \frac{\sqrt{3}}{4} (2\sqrt{3})^2 = \frac{\sqrt{3}}{4} \times 4 \times 3 = 3\sqrt{3} \text{ square units.} \]

### Step 4: Use the formula for the inradius.
The radius \( r \) of the incircle is given by the formula:
\[ r = \frac{A}{s} \]
Substituting the values of \( A \) and \( s \):
\[ r = \frac{3\sqrt{3}}{3\sqrt{3}} = 1 \text{ cm.} \]

### Conclusion:
The radius \( x \) of the incircle of the equilateral triangle is \( 1 \) cm.