Three vertices of a parallelogram ABCD are B (6,7) , C (8,3) and D `(0,-1)` . Find the co-ordinates of vertex A .

Answer

To find the coordinates of vertex A of the parallelogram ABCD, we will use the properties of the diagonals of a parallelogram, which bisect each other.

### Step-by-step Solution:

1. **Identify the given points**:
- B(6, 7)
- C(8, 3)
- D(0, -1)

2. **Let the coordinates of point A be (x, y)**.

3. **Find the midpoint O of diagonal BD**:
- The midpoint formula is given by:
\[ O_x = \frac{x_1 + x_2}{2}, \quad O_y = \frac{y_1 + y_2}{2} \]
- Here, we will find the midpoint O of points B and D:
\[ O_x = \frac{6 + 0}{2} = \frac{6}{2} = 3 \]
\[ O_y = \frac{7 + (-1)}{2} = \frac{6}{2} = 3 \]
- Therefore, the coordinates of point O are (3, 3).

4. **Since O is also the midpoint of diagonal AC**, we can set up the equations:
- For the x-coordinate:
\[ O_x = \frac{x + 8}{2} \]
Substituting the value of O_x:
\[ 3 = \frac{x + 8}{2} \]
Multiplying both sides by 2:
\[ 6 = x + 8 \]
Solving for x:
\[ x = 6 - 8 = -2 \]

5. **Now, find the y-coordinate**:
- For the y-coordinate:
\[ O_y = \frac{y + 3}{2} \]
Substituting the value of O_y:
\[ 3 = \frac{y + 3}{2} \]
Multiplying both sides by 2:
\[ 6 = y + 3 \]
Solving for y:
\[ y = 6 - 3 = 3 \]

6. **Conclusion**:
- The coordinates of point A are (-2, 3).

### Final Answer:
The coordinates of vertex A are (-2, 3).