The co-ordinates of three vertices of a parallelogram `ABCD` are `A(1,0)` , `B(3,4)` and `C(1,2)`. The co-ordinates of fourth vertex `D` are :

Options

A. `(-1,2)`
B. `(-5,-4)`
C. `(-1,-2)`
D. `(2,0)`

Answer

To find the coordinates of the fourth vertex \( D \) of the parallelogram \( ABCD \) given the coordinates of vertices \( A(1, 0) \), \( B(3, 4) \), and \( C(1, 2) \), we can use the properties of the parallelogram.

### Step-by-Step Solution:

1. **Identify the given points**:
- \( A(1, 0) \)
- \( B(3, 4) \)
- \( C(1, 2) \)
- Let the coordinates of point \( D \) be \( D(x, y) \).

2. **Use the property of midpoints**:
In a parallelogram, the diagonals bisect each other. Therefore, the midpoint of diagonal \( AC \) should be the same as the midpoint of diagonal \( BD \).

- **Calculate the midpoint of \( AC \)**:
\[ \text{Midpoint of } AC = \left( \frac{x_A + x_C}{2}, \frac{y_A + y_C}{2} \right) = \left( \frac{1 + 1}{2}, \frac{0 + 2}{2} \right) = \left( 1, 1 \right) \]

3. **Calculate the midpoint of \( BD \)**:
- The midpoint of \( BD \) is given by:
\[ \text{Midpoint of } BD = \left( \frac{x_B + x_D}{2}, \frac{y_B + y_D}{2} \right) = \left( \frac{3 + x}{2}, \frac{4 + y}{2} \right) \]

4. **Set the midpoints equal**:
Since the midpoints are equal, we can set the coordinates equal to each other:
\[ \frac{3 + x}{2} = 1 \quad \text{(1)} \]
\[ \frac{4 + y}{2} = 1 \quad \text{(2)} \]

5. **Solve equation (1)**:
\[ 3 + x = 2 \implies x = 2 - 3 \implies x = -1 \]

6. **Solve equation (2)**:
\[ 4 + y = 2 \implies y = 2 - 4 \implies y = -2 \]

7. **Conclusion**:
The coordinates of the fourth vertex \( D \) are:
\[ D(-1, -2) \]

### Final Answer:
The coordinates of the fourth vertex \( D \) are \( D(-1, -2) \).