If the centroid of triangle formed by point p(a, b), q(b, c) and r(c, a) is at the origin, what is the value of a+b+c?

Options

A. 3
B. 2
C. 1
D. 0

Answer

To solve the problem, we need to find the value of \( a + b + c \) given that the centroid of the triangle formed by the points \( P(a, b) \), \( Q(b, c) \), and \( R(c, a) \) is at the origin (0, 0).

### Step-by-Step Solution:

1. **Understanding the Centroid Formula**:
The centroid \( G \) of a triangle with vertices at \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]

2. **Identifying the Coordinates**:
For our triangle:
- \( P(a, b) \)
- \( Q(b, c) \)
- \( R(c, a) \)

Here, we have:
- \( x_1 = a \), \( y_1 = b \)
- \( x_2 = b \), \( y_2 = c \)
- \( x_3 = c \), \( y_3 = a \)

3. **Substituting into the Centroid Formula**:
The coordinates of the centroid can be calculated as follows:
\[ G\left(\frac{a + b + c}{3}, \frac{b + c + a}{3}\right) \]

4. **Setting the Centroid to the Origin**:
Since the centroid is at the origin, we set both coordinates to zero:
\[ \frac{a + b + c}{3} = 0 \quad \text{and} \quad \frac{b + c + a}{3} = 0 \]

5. **Solving the Equations**:
From the first equation:
\[ a + b + c = 0 \]

6. **Conclusion**:
Therefore, the value of \( a + b + c \) is:
\[ \boxed{0} \]