If in a triangle `ABC, r_(1)+r_(2)+r_(3)=9r`, then the triangle is necessarily

Options

A. right angled
B. equilateral
C. obtuse angled
D. none of these

Explanation

To solve the problem, we need to analyze the given condition in the triangle \( ABC \) where \( r_1 + r_2 + r_3 = 9r \). Here, \( r_1, r_2, r_3 \) are the exradii opposite to vertices \( A, B, \) and \( C \) respectively, and \( r \) is the inradius of triangle \( ABC \).

### Step-by-Step Solution:

1. **Understand the relationship between exradii and circumradius:**
The relationship between the exradii \( r_1, r_2, r_3 \) and the inradius \( r \) and circumradius \( R \) is given by:
\[ r_1 + r_2 + r_3 = 4R + r \]

2. **Substitute the given condition:**
According to the problem, we have:
\[ r_1 + r_2 + r_3 = 9r \]
We can set these two equations equal to each other:
\[ 4R + r = 9r \]

3. **Rearranging the equation:**
Rearranging the equation gives:
\[ 4R = 9r - r \]
Simplifying this, we find:
\[ 4R = 8r \]

4. **Solving for \( R \):**
Dividing both sides by 4, we get:
\[ R = 2r \]

5. **Identify the type of triangle:**
We know that in an equilateral triangle, the circumradius \( R \) is twice the inradius \( r \):
\[ R = 2r \]
This means that the triangle \( ABC \) must be equilateral.

### Conclusion:
Thus, if \( r_1 + r_2 + r_3 = 9r \), then the triangle \( ABC \) is necessarily an equilateral triangle.