`A(-4, 4), B(x, -1)` and C(6,y) are the vetices of `DeltaABC`. If the centroid of this triangle ABC is at the origin, find the values of x and y.

Answer

To find the values of \( x \) and \( y \) for the vertices \( A(-4, 4) \), \( B(x, -1) \), and \( C(6, y) \) such that the centroid of triangle \( ABC \) is at the origin \( (0, 0) \), we can use the formula for the centroid of a triangle.

### Step-by-Step Solution:

1. **Understanding the Centroid Formula**:
The centroid \( G \) of a triangle with vertices \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) is given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
For our triangle, the centroid is given as \( (0, 0) \).

2. **Setting Up the Equations**:
Using the coordinates of points \( A \), \( B \), and \( C \):
- \( A(-4, 4) \)
- \( B(x, -1) \)
- \( C(6, y) \)

We can set up the equations for the x-coordinate and y-coordinate of the centroid:
\[ \frac{-4 + x + 6}{3} = 0 \quad \text{(1)} \]
\[ \frac{4 - 1 + y}{3} = 0 \quad \text{(2)} \]

3. **Solving for \( x \)**:
From equation (1):
\[ \frac{-4 + x + 6}{3} = 0 \]
Simplifying this:
\[ -4 + x + 6 = 0 \]
\[ x + 2 = 0 \]
\[ x = -2 \]

4. **Solving for \( y \)**:
From equation (2):
\[ \frac{4 - 1 + y}{3} = 0 \]
Simplifying this:
\[ 4 - 1 + y = 0 \]
\[ 3 + y = 0 \]
\[ y = -3 \]

5. **Final Values**:
Therefore, the values of \( x \) and \( y \) are:
\[ x = -2, \quad y = -3 \]

### Summary of the Solution:
The values of \( x \) and \( y \) are:
- \( x = -2 \)
- \( y = -3 \)