If the sides of a triangle be 18, 24, 30 cms, then radius of the in-circle is

Options

A. 2
B. 4
C. 6
D. 9

Answer

To find the radius of the incircle of a triangle with sides 18 cm, 24 cm, and 30 cm, we can follow these steps:

### Step 1: Identify the sides of the triangle
Let the sides of the triangle be:
- \( a = 18 \) cm
- \( b = 24 \) cm
- \( c = 30 \) cm

### Step 2: Calculate the semi-perimeter (s)
The semi-perimeter \( s \) is given by the formula:
\[ s = \frac{a + b + c}{2} \]
Substituting the values:
\[ s = \frac{18 + 24 + 30}{2} = \frac{72}{2} = 36 \text{ cm} \]

### Step 3: Calculate the area of the triangle (Δ)
The area \( Δ \) can be calculated using Heron's formula:
\[ Δ = \sqrt{s(s - a)(s - b)(s - c)} \]
Substituting the values we have:
\[ Δ = \sqrt{36(36 - 18)(36 - 24)(36 - 30)} \]
Calculating each term:
\[ s - a = 36 - 18 = 18 \]
\[ s - b = 36 - 24 = 12 \]
\[ s - c = 36 - 30 = 6 \]
Now substituting these values into the area formula:
\[ Δ = \sqrt{36 \times 18 \times 12 \times 6} \]

### Step 4: Simplify the area calculation
To simplify \( Δ \):
\[ Δ = \sqrt{36} \times \sqrt{18} \times \sqrt{12} \times \sqrt{6} \]
Calculating \( \sqrt{36} = 6 \):
\[ Δ = 6 \times \sqrt{18 \times 12 \times 6} \]
Calculating \( 18 \times 12 = 216 \) and \( 216 \times 6 = 1296 \):
\[ Δ = 6 \times \sqrt{1296} = 6 \times 36 = 216 \text{ cm}^2 \]

### Step 5: Calculate the radius of the incircle (r)
The radius \( r \) of the incircle is given by the formula:
\[ r = \frac{Δ}{s} \]
Substituting the values we calculated:
\[ r = \frac{216}{36} = 6 \text{ cm} \]

### Final Answer
The radius of the incircle is \( 6 \) cm.

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