If the vertices of `DeltaABC` are A(a,b),B(b,c) and C(c,a) and its centrid is the origin, then the value of `(a+b+c)` is :

Options

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

Answer

To solve the problem, we need to find the value of \( a + b + c \) given that the centroid of triangle \( ABC \) with vertices \( A(a,b) \), \( B(b,c) \), and \( C(c,a) \) is the origin.

### Step-by-Step Solution:

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

2. **Substituting the Coordinates**:
For triangle \( ABC \):
- Vertex \( A \) has coordinates \( (a, b) \)
- Vertex \( B \) has coordinates \( (b, c) \)
- Vertex \( C \) has coordinates \( (c, a) \)

Therefore, the coordinates of the centroid \( G \) can be calculated as:
\[ G\left( \frac{a + b + c}{3}, \frac{b + c + a}{3} \right) \]

3. **Setting the Centroid to the Origin**:
Since the centroid is given to be the origin, we set:
\[ G\left( \frac{a + b + c}{3}, \frac{b + c + a}{3} \right) = (0, 0) \]

4. **Equating the Coordinates**:
This gives us two equations:
- From the x-coordinate:
\[ \frac{a + b + c}{3} = 0 \]
- From the y-coordinate:
\[ \frac{b + c + a}{3} = 0 \]

5. **Solving the Equations**:
From the first equation:
\[ a + b + c = 0 \]
This is the value we are looking for.

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