In a `DeltaABC` if `r_(1):r_(2):r_(3)=2:4:6,` then `a:b:c `=

Options

A. `3:5:7`
B. `1:2:3`
C. `5:8:7`
D. none of these

Explanation

To solve the problem, we need to find the ratio \( a:b:c \) given the ratio of the inradii \( r_1:r_2:r_3 = 2:4:6 \).

### Step-by-step Solution:

1. **Understanding the Relationship**:
The inradii \( r_1, r_2, r_3 \) correspond to the sides opposite to vertices \( A, B, C \) respectively. The relationship can be expressed as:
\[ \frac{r_1}{\Delta} = \frac{1}{s-a}, \quad \frac{r_2}{\Delta} = \frac{1}{s-b}, \quad \frac{r_3}{\Delta} = \frac{1}{s-c} \]
where \( \Delta \) is the area of triangle \( ABC \) and \( s \) is the semi-perimeter.

2. **Setting Up the Ratios**:
Given \( r_1:r_2:r_3 = 2:4:6 \), we can write:
\[ \frac{1}{s-a} : \frac{1}{s-b} : \frac{1}{s-c} = 2 : 4 : 6 \]

3. **Introducing a Constant**:
Let \( k \) be a constant such that:
\[ \frac{1}{s-a} = 2k, \quad \frac{1}{s-b} = 4k, \quad \frac{1}{s-c} = 6k \]

4. **Finding Expressions for \( s-a, s-b, s-c \)**:
From the above, we can express:
\[ s-a = \frac{1}{2k}, \quad s-b = \frac{1}{4k}, \quad s-c = \frac{1}{6k} \]

5. **Using the Semi-Perimeter**:
The semi-perimeter \( s \) is given by:
\[ s = a + b + c \]
We can express \( a, b, c \) in terms of \( s \):
\[ a = s - (s-a) = s - \frac{1}{2k}, \quad b = s - (s-b) = s - \frac{1}{4k}, \quad c = s - (s-c) = s - \frac{1}{6k} \]

6. **Substituting for \( s \)**:
Adding the expressions for \( s-a, s-b, s-c \):
\[ (s-a) + (s-b) + (s-c) = \frac{1}{2k} + \frac{1}{4k} + \frac{1}{6k} \]
To simplify, find a common denominator (which is \( 12k \)):
\[ \frac{6}{12k} + \frac{3}{12k} + \frac{2}{12k} = \frac{11}{12k} \]
Thus:
\[ 3s - (a + b + c) = \frac{11}{12k} \]
Since \( a + b + c = 2s \), we have:
\[ 3s - 2s = \frac{11}{12k} \implies s = \frac{11}{12k} \]

7. **Finding \( a, b, c \)**:
Substitute \( s \) back into the expressions for \( a, b, c \):
\[ a = s - \frac{1}{2k} = \frac{11}{12k} - \frac{6}{12k} = \frac{5}{12k} \]
\[ b = s - \frac{1}{4k} = \frac{11}{12k} - \frac{3}{12k} = \frac{8}{12k} \]
\[ c = s - \frac{1}{6k} = \frac{11}{12k} - \frac{2}{12k} = \frac{9}{12k} \]

8. **Finding the Ratio**:
The ratio \( a:b:c \) is:
\[ a:b:c = \frac{5}{12k} : \frac{8}{12k} : \frac{9}{12k} = 5:8:9 \]

### Final Result:
Thus, the ratio \( a:b:c = 5:8:9 \).