The radius of the incircle of a triangle whose sides are 9 cm, 12 cm and 15 cm is

Options

A. 9 cm
B. 13 cm
C. 3 cm
D. 6 cm

Answer

To find the radius of the incircle of a triangle with sides 9 cm, 12 cm, and 15 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
The sides of the triangle are given as:
- \( a = 9 \) cm
- \( b = 12 \) cm
- \( c = 15 \) cm

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) of the triangle is calculated using the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{9 + 12 + 15}{2} = \frac{36}{2} = 18 \text{ cm} \]

### Step 3: Calculate the area (Δ) of the triangle
Since the triangle is a right triangle (as \( 9^2 + 12^2 = 15^2 \)), we can use the formula for the area of a right triangle:
\[ \Delta = \frac{1}{2} \times \text{base} \times \text{height} \]
Here, we can take the base as 9 cm and the height as 12 cm:
\[ \Delta = \frac{1}{2} \times 9 \times 12 = \frac{108}{2} = 54 \text{ cm}^2 \]

### Step 4: Calculate the radius of the incircle (r)
The radius \( r \) of the incircle can be calculated using the formula:
\[ r = \frac{\Delta}{s} \]
Substituting the values we found:
\[ r = \frac{54}{18} = 3 \text{ cm} \]

### Conclusion
The radius of the incircle of the triangle is \( 3 \) cm.

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