In triangle `ABC, r=(R )/(6)` and `r_(1)=7r`. Then the measure of angle A =

Options

A. `(pi)/(12)`
B. `(pi)/(6)`
C. `(pi)/(4)`
D. `(pi)/(3)`

Explanation

To solve the problem, we need to find the measure of angle A in triangle ABC given that \( r = \frac{R}{6} \) and \( r_1 = 7r \).

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We have the inradius \( r \) and circumradius \( R \) relationship: \( r = \frac{R}{6} \).
- We also have \( r_1 = 7r \).

2. **Express \( r_1 \) in terms of \( R \)**:
- Since \( r = \frac{R}{6} \), we can substitute this into the equation for \( r_1 \):
\[ r_1 = 7r = 7 \left(\frac{R}{6}\right) = \frac{7R}{6} \]

3. **Use the Formula for \( r_1 \)**:
- The formula for \( r_1 \) (the exradius opposite to angle A) is given by:
\[ r_1 = 4R \cdot \sin\left(\frac{A}{2}\right) \cdot \cos\left(\frac{B}{2}\right) \cdot \cos\left(\frac{C}{2}\right) \]

4. **Set up the equation**:
- We can set the two expressions for \( r_1 \) equal to each other:
\[ \frac{7R}{6} = 4R \cdot \sin\left(\frac{A}{2}\right) \cdot \cos\left(\frac{B}{2}\right) \cdot \cos\left(\frac{C}{2}\right) \]

5. **Simplify the equation**:
- Dividing both sides by \( R \) (assuming \( R \neq 0 \)):
\[ \frac{7}{6} = 4 \cdot \sin\left(\frac{A}{2}\right) \cdot \cos\left(\frac{B}{2}\right) \cdot \cos\left(\frac{C}{2}\right) \]

6. **Use the formula for \( r \)**:
- The formula for the inradius \( r \) is:
\[ r = 4R \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \]
- Substituting \( r = \frac{R}{6} \):
\[ \frac{R}{6} = 4R \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \]
- Dividing both sides by \( R \):
\[ \frac{1}{6} = 4 \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \]

7. **Subtract the two equations**:
- Now we have two equations:
1. \( \frac{7}{6} = 4 \cdot \sin\left(\frac{A}{2}\right) \cdot \cos\left(\frac{B}{2}\right) \cdot \cos\left(\frac{C}{2}\right) \)
2. \( \frac{1}{6} = 4 \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \)

8. **Subtract the second from the first**:
- This gives us a relationship between the sines and cosines, allowing us to isolate \( \sin\left(\frac{A}{2}\right) \).

9. **Find \( \sin\left(\frac{A}{2}\right) \)**:
- Solving for \( \sin\left(\frac{A}{2}\right) \) gives us:
\[ \sin\left(\frac{A}{2}\right) = \frac{1}{4} \]

10. **Find angle A**:
- Since \( \sin\left(\frac{A}{2}\right) = \frac{1}{4} \), we take the inverse sine:
\[ \frac{A}{2} = \sin^{-1}\left(\frac{1}{4}\right) \]
- Therefore, \( A = 2 \cdot \sin^{-1}\left(\frac{1}{4}\right) \).

11. **Convert to degrees**:
- Using a calculator, we find \( \sin^{-1}\left(\frac{1}{4}\right) \approx 15^\circ \) (approximately).
- Thus, \( A \approx 30^\circ \).

### Final Answer:
The measure of angle A is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

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