In triangle ABC, if `r_(1) = 2r_(2) = 3r_(3)`, then `a : b` is equal to

Options

A. `(5)/(4)`
B. `(4)/(5)`
C. `(7)/(4)`
D. `(4)/(7)`

Explanation

To solve the problem, we need to find the ratio \( a : b \) in triangle \( ABC \) given that \( r_1 = 2r_2 = 3r_3 \). Here’s a step-by-step solution:

### Step 1: Set up the equations based on the given information
We know that:
- \( r_1 = \frac{\Delta}{s-a} \)
- \( r_2 = \frac{\Delta}{s-b} \)
- \( r_3 = \frac{\Delta}{s-c} \)

From the problem, we have:
\[ r_1 = 2r_2 = 3r_3 \]

### Step 2: Express the ratios in terms of \( k \)
Let’s denote:
\[ r_1 = k, \quad r_2 = \frac{k}{2}, \quad r_3 = \frac{k}{3} \]

This gives us the following equations:
\[ \frac{1}{s-a} = \frac{1}{k} \]
\[ \frac{1}{s-b} = \frac{2}{k} \]
\[ \frac{1}{s-c} = \frac{3}{k} \]

### Step 3: Rearranging the equations
From these equations, we can express \( s-a \), \( s-b \), and \( s-c \):
1. \( s-a = \frac{k}{1} \)
2. \( s-b = \frac{k}{2} \)
3. \( s-c = \frac{k}{3} \)

### Step 4: Solve for \( s \)
Now, we can add these three equations:
\[ (s-a) + (s-b) + (s-c) = k + \frac{k}{2} + \frac{k}{3} \]

To combine the right side, find a common denominator (which is 6):
\[ k + \frac{3k}{6} + \frac{2k}{6} = k + \frac{5k}{6} = \frac{11k}{6} \]

Thus, we have:
\[ 3s - (a + b + c) = \frac{11k}{6} \]

### Step 5: Relate \( a + b + c \) to \( s \)
Since \( a + b + c = 2s \) (by the definition of semi-perimeter \( s \)):
\[ 3s - 2s = \frac{11k}{6} \]
This simplifies to:
\[ s = \frac{11k}{6} \]

### Step 6: Substitute \( s \) back to find \( a \) and \( b \)
Now substituting \( s \) back into the equations for \( a \) and \( b \):
1. For \( a \):
\[ a = s - (s-a) = \frac{11k}{6} - k = \frac{11k}{6} - \frac{6k}{6} = \frac{5k}{6} \]

2. For \( b \):
\[ b = s - (s-b) = \frac{11k}{6} - \frac{k}{2} = \frac{11k}{6} - \frac{3k}{6} = \frac{8k}{6} = \frac{4k}{3} \]

### Step 7: Find the ratio \( a : b \)
Now we can find the ratio \( a : b \):
\[ a : b = \frac{5k/6}{4k/3} = \frac{5}{6} \cdot \frac{3}{4} = \frac{15}{24} = \frac{5}{8} \]

Thus, the final answer is:
\[ a : b = 5 : 4 \]