If origin is the centroid of a triangle ABC having vertices A(a,1,3), B(-2,b,-5) and C(4,7, c) , then the values of a, b, c are

Options

A. a = - 2, b = 8 , c = 2
B. a = 2, b = 8, c = - 2
C. a = 2, b = - 8, c = 2
D. a = - 2, b = - 8, c = 2

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) given that the origin (0, 0, 0) is the centroid of triangle ABC with vertices \( A(a, 1, 3) \), \( B(-2, b, -5) \), and \( C(4, 7, c) \).

### Step-by-Step Solution:

1. **Understanding the Centroid Formula**:
The centroid \( G \) of a triangle with vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) is given by:
\[ G = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

2. **Substituting the Coordinates**:
For our triangle, we substitute the coordinates of points A, B, and C:
\[ G = \left( \frac{a + (-2) + 4}{3}, \frac{1 + b + 7}{3}, \frac{3 + (-5) + c}{3} \right) \]

3. **Simplifying the Centroid Coordinates**:
Simplifying the expressions for the centroid, we get:
\[ G = \left( \frac{a + 2}{3}, \frac{b + 8}{3}, \frac{c - 2}{3} \right) \]

4. **Setting the Centroid Equal to the Origin**:
Since the centroid is given to be the origin (0, 0, 0), we can set the coordinates equal to zero:
\[ \frac{a + 2}{3} = 0, \quad \frac{b + 8}{3} = 0, \quad \frac{c - 2}{3} = 0 \]

5. **Solving for \( a \), \( b \), and \( c \)**:
- From \( \frac{a + 2}{3} = 0 \):
\[ a + 2 = 0 \implies a = -2 \]
- From \( \frac{b + 8}{3} = 0 \):
\[ b + 8 = 0 \implies b = -8 \]
- From \( \frac{c - 2}{3} = 0 \):
\[ c - 2 = 0 \implies c = 2 \]

6. **Final Values**:
Thus, the values of \( a \), \( b \), and \( c \) are:
\[ a = -2, \quad b = -8, \quad c = 2 \]

### Summary of Results:
- \( a = -2 \)
- \( b = -8 \)
- \( c = 2 \)