Given that `Delta = 6, r_(1) = 3, r_(3) = 6`
Circumradius R is equal to

Options

A. 2.5
B. 3.5
C. 1.5
D. none of these

Explanation

To find the circumradius \( R \) of the triangle given the area \( \Delta = 6 \), \( r_1 = 3 \), and \( r_3 = 6 \), we will follow these steps:

### Step 1: Use the relationship between the inradii and semi-perimeter
We know that:
\[ r_1 = \frac{\Delta}{s-a}, \quad r_2 = \frac{\Delta}{s-b}, \quad r_3 = \frac{\Delta}{s-c} \]
where \( s \) is the semi-perimeter of the triangle and \( a, b, c \) are the sides opposite to angles \( A, B, C \) respectively.

### Step 2: Set up the equations
From the given values:
1. \( r_1 = 3 \) implies \( 3 = \frac{6}{s-a} \) → \( s-a = 2 \) → \( a = s - 2 \)
2. \( r_3 = 6 \) implies \( 6 = \frac{6}{s-c} \) → \( s-c = 1 \) → \( c = s - 1 \)

### Step 3: Express \( b \) in terms of \( s \)
Since we have \( r_2 \) not provided, we can find \( b \) using the semi-perimeter:
\[ s = \frac{a + b + c}{2} \Rightarrow b = 2s - a - c \]
Substituting \( a \) and \( c \):
\[ b = 2s - (s - 2) - (s - 1) = 2s - s + 2 - s + 1 = 3 \]

### Step 4: Determine the values of \( a, b, c \)
From the equations:
- \( s - a = 2 \) → \( a = s - 2 \)
- \( s - c = 1 \) → \( c = s - 1 \)
- \( b = 3 \)

Now we can express \( s \):
\[ s = a + b + c = (s - 2) + 3 + (s - 1) = 2s \]
This gives us \( s = 6 \).

Now substituting \( s \) back:
- \( a = s - 2 = 6 - 2 = 4 \)
- \( b = 3 \)
- \( c = s - 1 = 6 - 1 = 5 \)

### Step 5: Calculate the circumradius \( R \)
The circumradius \( R \) is given by:
\[ R = \frac{abc}{4\Delta} \]
Substituting the values:
\[ R = \frac{4 \times 3 \times 5}{4 \times 6} = \frac{60}{24} = \frac{5}{2} \]

### Final Answer:
Thus, the circumradius \( R \) is:
\[ R = 2.5 \]