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 are (7, - 1) and (1, 2) respectively ?

Options

A. `(-2, 5)`
B. `(2,5)`
C. `(9-2,-5)`
D. `(2,-5)`

Answer

To find the coordinates of vertex C of triangle ABC, given the coordinates of the centroid and the other two vertices, 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:
\[ G\left(\frac{x1 + x2 + x3}{3}, \frac{y1 + y2 + y3}{3}\right) \]
In this case, the centroid G is given as (2, 2).

2. **Assign Known Values**:
- Coordinates of vertex A (x1, y1) = (7, -1)
- Coordinates of vertex B (x2, y2) = (1, 2)
- Coordinates of vertex C (x3, y3) = (x, y) (unknown)

3. **Set Up the Equations**:
From the centroid formula, we can set up two equations:
\[ \frac{x1 + x2 + x3}{3} = 2 \]
\[ \frac{y1 + y2 + y3}{3} = 2 \]

4. **Substitute Known Values into the X-Coordinate Equation**:
Substitute the values of A and B into the first equation:
\[ \frac{7 + 1 + x}{3} = 2 \]
Multiply both sides by 3:
\[ 7 + 1 + x = 6 \]
Simplify:
\[ 8 + x = 6 \]
Solve for x:
\[ x = 6 - 8 = -2 \]

5. **Substitute Known Values into the Y-Coordinate Equation**:
Substitute the values of A and B into the second equation:
\[ \frac{-1 + 2 + y}{3} = 2 \]
Multiply both sides by 3:
\[ -1 + 2 + y = 6 \]
Simplify:
\[ 1 + y = 6 \]
Solve for y:
\[ y = 6 - 1 = 5 \]

6. **Conclusion**:
The coordinates of vertex C are:
\[ C(-2, 5) \]

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