If (-2,-1), (1,0) and (4,3) are three successive vertices of a parallelogram, find the fourth vertex.

Answer

To find the fourth vertex of the parallelogram given the three successive vertices (-2, -1), (1, 0), and (4, 3), we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Label the vertices**:
Let the vertices be labeled as follows:
- A = (-2, -1)
- B = (1, 0)
- C = (4, 3)
- D = (x, y) (the vertex we need to find)

2. **Find the midpoint of diagonal AC**:
The midpoint M of diagonal AC can be calculated using the midpoint formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
Here, A = (-2, -1) and C = (4, 3).
\[ M = \left( \frac{-2 + 4}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{2}{2}, \frac{2}{2} \right) = (1, 1) \]

3. **Set up the equation for midpoint of diagonal BD**:
The midpoint of diagonal BD must also equal M. Thus, we can express the midpoint of BD as:
\[ M = \left( \frac{1 + x}{2}, \frac{0 + y}{2} \right) \]
Setting this equal to (1, 1):
\[ \frac{1 + x}{2} = 1 \quad \text{and} \quad \frac{0 + y}{2} = 1 \]

4. **Solve for x and y**:
From the first equation:
\[ 1 + x = 2 \implies x = 2 \]
From the second equation:
\[ y = 2 \]

5. **Conclusion**:
Therefore, the coordinates of the fourth vertex D are (2, 2).

### Final Answer:
The fourth vertex of the parallelogram is **(2, 2)**.