If the origin is the centroid of a triangle ABC having vertices `A(a ,1,3),\ B(-2, b ,-5)a n d\ C(4,7, c)` , find the values of `a , b , c dot`

Explanation

To find the values of \( a \), \( b \), and \( c \) such that the origin is the centroid of triangle \( ABC \) with vertices \( A(a, 1, 3) \), \( B(-2, b, -5) \), and \( C(4, 7, c) \), we can follow these steps:

### Step 1: Understand the formula for the centroid
The coordinates of the centroid \( G \) of a triangle with vertices \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) are given by:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]

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