The co - ordinates of the centroid of a triangle ABC are (2,2) what are the co ordinates of vertex c if co ordinates of a and b (7,-1) and (1,2) respectively

Options

A. `(-2,5)`
B. `(2,5)`
C. `(-2,-5)`
D. `(2,-5)`

Answer

To find the coordinates of vertex C of triangle ABC given the coordinates of vertices A and B, and the centroid of the triangle, 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:
\[ G\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]

2. **Identify Given Values**:
We are given:
- Centroid G = (2, 2)
- Vertex A = (7, -1)
- Vertex B = (1, 2)
- Vertex C = (x, y) (unknown)

3. **Set Up the Equation for X-Coordinates**:
From the centroid formula for the x-coordinates:
\[ \frac{x_A + x_B + x_C}{3} = 2 \]
Substituting the known values:
\[ \frac{7 + 1 + x}{3} = 2 \]

4. **Solve for x**:
Multiply both sides by 3:
\[ 7 + 1 + x = 6 \]
Simplifying gives:
\[ 8 + x = 6 \]
Subtract 8 from both sides:
\[ x = 6 - 8 = -2 \]

5. **Set Up the Equation for Y-Coordinates**:
Now, for the y-coordinates:
\[ \frac{y_A + y_B + y_C}{3} = 2 \]
Substituting the known values:
\[ \frac{-1 + 2 + y}{3} = 2 \]

6. **Solve for y**:
Multiply both sides by 3:
\[ -1 + 2 + y = 6 \]
Simplifying gives:
\[ 1 + y = 6 \]
Subtract 1 from both sides:
\[ y = 6 - 1 = 5 \]

7. **Final Coordinates of Vertex C**:
Therefore, the coordinates of vertex C are:
\[ C(-2, 5) \]

### Conclusion:
The coordinates of vertex C are (-2, 5).