If the points `A(1,-2),B(2,3),C(a,2)andD(-4,-3)` forms a parallelogram, find the value of 'a'.

Answer

To find the value of 'a' such that the points A(1, -2), B(2, 3), C(a, 2), and D(-4, -3) form a parallelogram, we can use the property that the midpoints of the diagonals of a parallelogram are the same.

### Step-by-Step Solution:

1. **Identify the Midpoints of the Diagonals:**
- The diagonals of the parallelogram are AC and BD.
- The midpoint of diagonal AC can be calculated using the formula:
\[ \text{Midpoint of AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
- For points A(1, -2) and C(a, 2):
\[ \text{Midpoint of AC} = \left( \frac{1 + a}{2}, \frac{-2 + 2}{2} \right) = \left( \frac{1 + a}{2}, 0 \right) \]

2. **Calculate the Midpoint of Diagonal BD:**
- For points B(2, 3) and D(-4, -3):
\[ \text{Midpoint of BD} = \left( \frac{2 + (-4)}{2}, \frac{3 + (-3)}{2} \right) = \left( \frac{2 - 4}{2}, \frac{3 - 3}{2} \right) = \left( -1, 0 \right) \]

3. **Set the Midpoints Equal:**
- Since the midpoints of the diagonals must be equal, we can set the x-coordinates equal to each other:
\[ \frac{1 + a}{2} = -1 \]

4. **Solve for 'a':**
- Multiply both sides by 2 to eliminate the fraction:
\[ 1 + a = -2 \]
- Subtract 1 from both sides:
\[ a = -2 - 1 = -3 \]

5. **Conclusion:**
- The value of 'a' is -3.

### Final Answer:
The value of 'a' is -3.