If `R_(1)` is the circumradius of the pedal triangle of a given triangle ABC, and `R_(2)` is the circumradius of the pedal triangle of the pedal triangle formed, and so on `R_(3), R_(4)`..., then the value of `sum_( i=1)^(oo) R_(i)`, where R (circumradius) of `DeltaABC` is 5 is

Options

A. 8
B. 10
C. 12
D. 15

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the circumradius of triangle ABC
Given that the circumradius \( R \) of triangle ABC is 5.

### Step 2: Determine the circumradius of the pedal triangle \( R_1 \)
The circumradius of the pedal triangle \( R_1 \) is half of the circumradius of triangle ABC. Therefore:
\[ R_1 = \frac{R}{2} = \frac{5}{2} \]

### Step 3: Determine the circumradius of the next pedal triangle \( R_2 \)
The circumradius of the pedal triangle of the pedal triangle \( R_2 \) is also half of \( R_1 \):
\[ R_2 = \frac{R_1}{2} = \frac{5/2}{2} = \frac{5}{4} \]

### Step 4: Determine the circumradius of the next pedal triangle \( R_3 \)
Continuing this pattern, the circumradius of the pedal triangle of \( R_2 \) is:
\[ R_3 = \frac{R_2}{2} = \frac{5/4}{2} = \frac{5}{8} \]

### Step 5: Generalize the circumradius for \( R_i \)
From the pattern, we can see that:
\[ R_i = \frac{5}{2^{i-1}} \quad \text{for } i = 1, 2, 3, \ldots \]

### Step 6: Set up the summation
We need to find the sum:
\[ \sum_{i=1}^{\infty} R_i = R_1 + R_2 + R_3 + \ldots \]
Substituting the values we found:
\[ \sum_{i=1}^{\infty} R_i = 5 + \frac{5}{2} + \frac{5}{4} + \frac{5}{8} + \ldots \]

### Step 7: Recognize the series as a geometric series
This is a geometric series where the first term \( a = 5 \) and the common ratio \( r = \frac{1}{2} \):
\[ \sum_{i=1}^{\infty} R_i = 5 \left(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\right) \]

### Step 8: Calculate the sum of the infinite geometric series
The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Here, \( a = 1 \) and \( r = \frac{1}{2} \):
\[ S = \frac{1}{1 - \frac{1}{2}} = 2 \]

### Step 9: Multiply by the first term
Now, substituting back:
\[ \sum_{i=1}^{\infty} R_i = 5 \times 2 = 10 \]

### Final Answer
Thus, the value of \( \sum_{i=1}^{\infty} R_i \) is \( 10 \).