Two vertices of a `DeltaABC` are given by `A(6,4)and B(-2,2)` and its centroid is G (3,4). Find the co-ordinates of the vertex C of `DeltaABC.`

Answer

To find the coordinates of vertex C of triangle ABC given the vertices A(6, 4), B(-2, 2), and the centroid G(3, 4), we can use the formula for the centroid of a triangle.

### Step-by-step Solution:

1. **Understand the Centroid Formula**:
The coordinates of the centroid G of triangle ABC can be calculated using the formula:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
where \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are the coordinates of vertices A, B, and C respectively.

2. **Assign Known Values**:
We know:
- A(6, 4) → \( x_1 = 6 \), \( y_1 = 4 \)
- B(-2, 2) → \( x_2 = -2 \), \( y_2 = 2 \)
- G(3, 4) → \( G_x = 3 \), \( G_y = 4 \)
- Let C be \( (x_3, y_3) \).

3. **Set Up the Equations**:
Using the centroid formula, we can set up two equations:
\[ \frac{6 + (-2) + x_3}{3} = 3 \]
\[ \frac{4 + 2 + y_3}{3} = 4 \]

4. **Solve for \( x_3 \)**:
From the first equation:
\[ \frac{6 - 2 + x_3}{3} = 3 \]
Simplifying:
\[ \frac{4 + x_3}{3} = 3 \]
Multiply both sides by 3:
\[ 4 + x_3 = 9 \]
Subtract 4 from both sides:
\[ x_3 = 5 \]

5. **Solve for \( y_3 \)**:
From the second equation:
\[ \frac{4 + 2 + y_3}{3} = 4 \]
Simplifying:
\[ \frac{6 + y_3}{3} = 4 \]
Multiply both sides by 3:
\[ 6 + y_3 = 12 \]
Subtract 6 from both sides:
\[ y_3 = 6 \]

6. **Conclusion**:
Therefore, the coordinates of vertex C are:
\[ C(5, 6) \]