If A( 1,2) , B( 4,3) and C ( 0,0) are three vertices of parallelogram ABCD , find the coordinates of D.

Answer

To find the coordinates of point D in the parallelogram ABCD, where A(1, 2), B(4, 3), and C(0, 0) are given, 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 = (1, 2)
- B = (4, 3)
- C = (0, 0)

2. **Use the midpoint formula to find the midpoint of AC:**
The midpoint M of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
For points A(1, 2) and C(0, 0):
\[ M_{AC} = \left( \frac{1 + 0}{2}, \frac{2 + 0}{2} \right) = \left( \frac{1}{2}, 1 \right) \]

3. **Let the coordinates of point D be (x, y). Find the midpoint of BD:**
Using the same midpoint formula for points B(4, 3) and D(x, y):
\[ M_{BD} = \left( \frac{4 + x}{2}, \frac{3 + y}{2} \right) \]

4. **Set the midpoints equal to each other:**
Since the midpoints of diagonals AC and BD are equal, we can set them equal:
\[ \left( \frac{4 + x}{2}, \frac{3 + y}{2} \right) = \left( \frac{1}{2}, 1 \right) \]

5. **Equate the x-coordinates:**
\[ \frac{4 + x}{2} = \frac{1}{2} \]
Multiply both sides by 2:
\[ 4 + x = 1 \]
Solving for x:
\[ x = 1 - 4 = -3 \]

6. **Equate the y-coordinates:**
\[ \frac{3 + y}{2} = 1 \]
Multiply both sides by 2:
\[ 3 + y = 2 \]
Solving for y:
\[ y = 2 - 3 = -1 \]

7. **Conclusion:**
The coordinates of point D are:
\[ D = (-3, -1) \]

### Final Answer:
The coordinates of point D are (-3, -1).