Let a, b and c be the length of sides of a triangle ABC such that `(a+b)/7 = (b+c)/(8) = (c+a)/9` If r and R are the radius of incircle and radius of circumcircle of the triangle ABC, respectively, then the value of `R/r` s equal to :

Options

A. `5/2`
B. 2
C. `3/2`
D. 1

Answer

To solve the problem, we need to find the value of \( \frac{R}{r} \) where \( R \) is the circumradius and \( r \) is the inradius of triangle \( ABC \) with sides \( a, b, c \) satisfying the given conditions.

Let's break down the solution step by step:

### Step 1: Set up the equations
Given the condition:
\[ \frac{a+b}{7} = \frac{b+c}{8} = \frac{c+a}{9} = k \]
From this, we can express \( a+b, b+c, \) and \( c+a \) in terms of \( k \):
\[ a + b = 7k \quad (1) \]
\[ b + c = 8k \quad (2) \]
\[ c + a = 9k \quad (3) \]

### Step 2: Add the equations
Now, we add equations (1), (2), and (3):
\[ (a+b) + (b+c) + (c+a) = 7k + 8k + 9k \]
This simplifies to:
\[ 2(a + b + c) = 24k \]
Thus, we find:
\[ a + b + c = 12k \quad (4) \]

### Step 3: Solve for individual sides
Now we can solve for each side using equations (1), (2), and (3):
- From (1): \( c = 12k - (a + b) = 12k - 7k = 5k \)
- From (2): \( a = 12k - (b + c) = 12k - 8k = 4k \)
- From (3): \( b = 12k - (c + a) = 12k - 9k = 3k \)

Thus, we have:
\[ a = 4k, \quad b = 3k, \quad c = 5k \]

### Step 4: Calculate the circumradius \( R \)
The circumradius \( R \) of a triangle can be calculated using the formula:
\[ R = \frac{abc}{4\Delta} \]
where \( \Delta \) is the area of the triangle.

First, we calculate the area \( \Delta \):
Using Heron's formula, we first find the semi-perimeter \( s \):
\[ s = \frac{a + b + c}{2} = \frac{12k}{2} = 6k \]
Now, the area \( \Delta \) can be calculated as:
\[ \Delta = \sqrt{s(s-a)(s-b)(s-c)} \]
Substituting the values:
\[ \Delta = \sqrt{6k(6k - 4k)(6k - 3k)(6k - 5k)} = \sqrt{6k \cdot 2k \cdot 3k \cdot k} = \sqrt{36k^4} = 6k^2 \]

Now substituting \( a, b, c \):
\[ abc = (4k)(3k)(5k) = 60k^3 \]
Thus, substituting into the formula for \( R \):
\[ R = \frac{60k^3}{4 \cdot 6k^2} = \frac{60k^3}{24k^2} = \frac{5k}{2} \]

### Step 5: Calculate the inradius \( r \)
The inradius \( r \) can be calculated using the formula:
\[ r = \frac{\Delta}{s} \]
Substituting the values:
\[ r = \frac{6k^2}{6k} = k \]

### Step 6: Calculate \( \frac{R}{r} \)
Now we can find \( \frac{R}{r} \):
\[ \frac{R}{r} = \frac{\frac{5k}{2}}{k} = \frac{5}{2} \]

### Final Answer
Thus, the value of \( \frac{R}{r} \) is:
\[ \frac{R}{r} = \frac{5}{2} \]