The radius of the incircle of a triangle is 2 cm. If the area of the triangle is `6 cm^(2)`, then its perimeter is

Options

A. 2 cm
B. 3 cm
C. 6 cm
D. 9 cm

Answer

To find the perimeter of a triangle given the radius of its incircle (r) and its area (A), we can use the formula that relates these quantities:

\[ A = r \times s \]

where \( s \) is the semi-perimeter of the triangle. The semi-perimeter \( s \) can be expressed in terms of the perimeter \( P \) as:

\[ s = \frac{P}{2} \]

### Step 1: Substitute the known values into the area formula
Given:
- Radius of incircle \( r = 2 \, \text{cm} \)
- Area of triangle \( A = 6 \, \text{cm}^2 \)

Using the area formula:

\[ A = r \times s \]

Substituting the known values:

\[ 6 = 2 \times s \]

### Step 2: Solve for the semi-perimeter \( s \)
To find \( s \), we can rearrange the equation:

\[ s = \frac{6}{2} = 3 \, \text{cm} \]

### Step 3: Calculate the perimeter \( P \)
Since we know that \( s = \frac{P}{2} \), we can express the perimeter \( P \) as:

\[ P = 2s \]

Substituting the value of \( s \):

\[ P = 2 \times 3 = 6 \, \text{cm} \]

### Final Answer
The perimeter of the triangle is \( 6 \, \text{cm} \).

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