If a,b and c are the position vectors of the vertices A,B and C of the `DeltaABC`, then the centroid of `DeltaABC` is

Options

A. A. `(a+b+c)/(3)`
B. B. `(1)/(2)(a+(b+c)/(2))`
C. C. `a+(b+c)/(2)`
D. D. `(a+b+c)/(2)`

Answer

To find the centroid of triangle ABC with position vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) for vertices A, B, and C respectively, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Centroid**:
The centroid (G) of a triangle is the point where the three medians intersect. It divides each median in the ratio 2:1.

2. **Identify the Position Vectors**:
Let:
- \( \mathbf{A} \) be the position vector of vertex A, represented by \( \mathbf{a} \).
- \( \mathbf{B} \) be the position vector of vertex B, represented by \( \mathbf{b} \).
- \( \mathbf{C} \) be the position vector of vertex C, represented by \( \mathbf{c} \).

3. **Determine the Midpoint of Side BC**:
The midpoint D of side BC can be calculated using the midpoint formula:
\[ \mathbf{D} = \frac{\mathbf{b} + \mathbf{c}}{2} \]

4. **Apply the Internal Section Formula**:
Since G divides the median AD in the ratio 2:1, we can use the internal section formula to find the position vector of G:
\[ \mathbf{G} = \frac{m \cdot \mathbf{D} + n \cdot \mathbf{A}}{m+n} \]
Here, \( m = 2 \) (the part closer to D) and \( n = 1 \) (the part closer to A):
\[ \mathbf{G} = \frac{2 \cdot \left( \frac{\mathbf{b} + \mathbf{c}}{2} \right) + 1 \cdot \mathbf{a}}{2 + 1} \]

5. **Simplify the Expression**:
Substitute \( \mathbf{D} \) into the equation:
\[ \mathbf{G} = \frac{2 \cdot \frac{\mathbf{b} + \mathbf{c}}{2} + \mathbf{a}}{3} \]
This simplifies to:
\[ \mathbf{G} = \frac{\mathbf{b} + \mathbf{c} + \mathbf{a}}{3} \]

6. **Final Result**:
Thus, the position vector of the centroid G of triangle ABC is:
\[ \mathbf{G} = \frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{3} \]

### Conclusion:
The centroid of triangle ABC is given by the position vector:
\[ \mathbf{G} = \frac{\mathbf{a} + \mathbf{b} + \mathbf{c}}{3} \]