The inradius of triangle is 4 cm and its area is 34 sq. cm. the perimeter of the triangle is:

Options

A. 8.5 cm
B. 17 cm
C. 34 cm
D. 20 cm

Answer

To solve the problem step by step, we will use the relationship between the inradius, area, semi-perimeter, and perimeter of a triangle.

### Step-by-Step Solution:

1. **Understand the relationship**: The formula relating the inradius (r), area (A), and semi-perimeter (s) of a triangle is given by:
\[ r = \frac{A}{s} \]
where \( r \) is the inradius, \( A \) is the area, and \( s \) is the semi-perimeter.

2. **Substitute the given values**: We know from the problem that the inradius \( r = 4 \) cm and the area \( A = 34 \) cm². We can substitute these values into the formula:
\[ 4 = \frac{34}{s} \]

3. **Solve for the semi-perimeter (s)**: Rearranging the equation to solve for \( s \), we get:
\[ s = \frac{34}{4} \]
Simplifying this gives:
\[ s = \frac{34}{4} = \frac{17}{2} \text{ cm} \]

4. **Calculate the perimeter (P)**: The perimeter \( P \) of the triangle is twice the semi-perimeter:
\[ P = 2s \]
Substituting the value of \( s \):
\[ P = 2 \times \frac{17}{2} = 17 \text{ cm} \]

5. **Final answer**: Therefore, the perimeter of the triangle is:
\[ \boxed{17 \text{ cm}} \]