A,B,C are the points (-2,-1),(0,3),(4,0). Then the co ordinates of the point D such that ABCD is parallelogram are……

Answer

To find the coordinates of point D such that ABCD forms a parallelogram, we can use the property that the diagonals of a parallelogram bisect each other.

### Step-by-Step Solution:

1. **Identify the Coordinates of Points A, B, and C:**
- A = (-2, -1)
- B = (0, 3)
- C = (4, 0)

2. **Calculate the Midpoint of Diagonal AC:**
The midpoint M of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x1 + x2}{2}, \frac{y1 + y2}{2} \right) \]
For points A and C:
\[ M_{AC} = \left( \frac{-2 + 4}{2}, \frac{-1 + 0}{2} \right) = \left( \frac{2}{2}, \frac{-1}{2} \right) = (1, -\frac{1}{2}) \]

3. **Let the Coordinates of Point D be (x, y):**
We need to find the coordinates of D such that the midpoint of diagonal BD is the same as the midpoint of diagonal AC.

4. **Calculate the Midpoint of Diagonal BD:**
The midpoint M of points B and D is:
\[ M_{BD} = \left( \frac{0 + x}{2}, \frac{3 + y}{2} \right) \]

5. **Set the Midpoints Equal:**
Since M_{AC} = M_{BD}, we have:
\[ \left( \frac{0 + x}{2}, \frac{3 + y}{2} \right) = (1, -\frac{1}{2}) \]

This gives us two equations:
- From the x-coordinates:
\[ \frac{x}{2} = 1 \implies x = 2 \]
- From the y-coordinates:
\[ \frac{3 + y}{2} = -\frac{1}{2} \implies 3 + y = -1 \implies y = -1 - 3 = -4 \]

6. **Conclusion:**
Therefore, the coordinates of point D are:
\[ D = (2, -4) \]

### Final Answer:
The coordinates of point D such that ABCD is a parallelogram are **(2, -4)**.