If (1,-3) , (-2, 3) and (-2, 2) are three vertices of a parallelogram taken in that order , then the fourth vertex is `"_______"`

Options

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

Explanation

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

Let's denote the fourth vertex as D(a, b).

### Step-by-step Solution:

1. **Identify the Coordinates of the Given Vertices:**
- A = (1, -3)
- B = (-2, 3)
- C = (-2, 2)

2. **Use the Midpoint Formula:**
The midpoint of the diagonal AC should be equal to the midpoint of the diagonal BD. The midpoint formula for two points (x1, y1) and (x2, y2) is given by:
\[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

3. **Calculate the Midpoint of AC:**
For points A(1, -3) and C(-2, 2):
\[ \text{Midpoint of AC} = \left(\frac{1 + (-2)}{2}, \frac{-3 + 2}{2}\right) = \left(\frac{-1}{2}, \frac{-1}{2}\right) \]

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

5. **Set the Midpoints Equal:**
Since the midpoints of AC and BD are equal:
\[ \left(\frac{-2 + a}{2}, \frac{3 + b}{2}\right) = \left(\frac{-1}{2}, \frac{-1}{2}\right) \]

6. **Set Up the Equations:**
From the x-coordinates:
\[ \frac{-2 + a}{2} = \frac{-1}{2} \]
From the y-coordinates:
\[ \frac{3 + b}{2} = \frac{-1}{2} \]

7. **Solve for a:**
Multiply both sides of the first equation by 2:
\[ -2 + a = -1 \implies a = -1 + 2 \implies a = 1 \]

8. **Solve for b:**
Multiply both sides of the second equation by 2:
\[ 3 + b = -1 \implies b = -1 - 3 \implies b = -4 \]

9. **Conclusion:**
The coordinates of the fourth vertex D are (1, -4).

### Final Answer:
The fourth vertex is **(1, -4)**.