If in ` A B C ,A=pi/7,B=(2pi)/7,C=(4pi)/7` then `a^2+b^2+c^2` must be `R^2` (b) `3R^2` (c) `4R^2` (d) `7R^2`

Options

A. `R^(2)`
B. `3R^(2)`
C. `4R^(2)`
D. `7R^(2)`

Explanation

To solve the problem, we need to find the value of \( a^2 + b^2 + c^2 \) given the angles \( A = \frac{\pi}{7}, B = \frac{2\pi}{7}, C = \frac{4\pi}{7} \). We will use the relationship between the sides of the triangle and the sine of the angles.

### Step-by-Step Solution:

1. **Identify the Relationship**:
We know that in a triangle, the sides opposite to angles \( A, B, C \) can be expressed in terms of the circumradius \( R \) as follows:
\[
a = 2R \sin A, \quad b = 2R \sin B, \quad c = 2R \sin C
...