In a `DeltaABC` if `r_(1)=16,r_(2)=48` and `r_(3)=24`, then its in-radius ,is

Options

A. 7
B. 8
C. 6
D. none of these

Answer

To find the in-radius \( r \) of triangle \( ABC \) given \( r_1 = 16 \), \( r_2 = 48 \), and \( r_3 = 24 \), we can use the formula:

\[ \frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} \]

### Step-by-step Solution:

1. **Substitute the values of \( r_1 \), \( r_2 \), and \( r_3 \)**:
\[ \frac{1}{r} = \frac{1}{16} + \frac{1}{48} + \frac{1}{24} \]

2. **Find a common denominator**:
The least common multiple (LCM) of \( 16 \), \( 48 \), and \( 24 \) is \( 48 \).

3. **Rewrite each fraction with the common denominator**:
\[ \frac{1}{16} = \frac{3}{48}, \quad \frac{1}{48} = \frac{1}{48}, \quad \frac{1}{24} = \frac{2}{48} \]

4. **Add the fractions**:
\[ \frac{1}{r} = \frac{3}{48} + \frac{1}{48} + \frac{2}{48} = \frac{3 + 1 + 2}{48} = \frac{6}{48} \]

5. **Simplify the fraction**:
\[ \frac{6}{48} = \frac{1}{8} \]

6. **Take the reciprocal to find \( r \)**:
\[ r = 8 \]

### Final Answer:
The in-radius \( r \) of triangle \( ABC \) is \( 8 \).

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