A (4,8), B(-2,6) are two vertices of a triangle ABC. Find the coordinates of C if the centroid of the triangle is (2,7).

Answer

To find the coordinates of point C given the vertices A(4,8), B(-2,6), and the centroid G(2,7) of triangle ABC, 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 a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3) are given by:
\[ G\left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \]
Here, we know G(2,7), A(4,8), and B(-2,6). We need to find C(x, y).

2. **Set Up the Equations**:
Using the centroid formula, we can set up two equations:
- For the x-coordinates:
\[ \frac{4 + (-2) + x}{3} = 2 \]
- For the y-coordinates:
\[ \frac{8 + 6 + y}{3} = 7 \]

3. **Solve for x**:
- Start with the x-coordinate equation:
\[ \frac{4 - 2 + x}{3} = 2 \]
- Simplify:
\[ \frac{2 + x}{3} = 2 \]
- Multiply both sides by 3:
\[ 2 + x = 6 \]
- Subtract 2 from both sides:
\[ x = 4 \]

4. **Solve for y**:
- Now, use the y-coordinate equation:
\[ \frac{8 + 6 + y}{3} = 7 \]
- Simplify:
\[ \frac{14 + y}{3} = 7 \]
- Multiply both sides by 3:
\[ 14 + y = 21 \]
- Subtract 14 from both sides:
\[ y = 7 \]

5. **Conclusion**:
The coordinates of point C are (4, 7).

### Final Answer:
C(4, 7)