The three vertices of a parallelogram taken in a order are (-1,0) , (3,1) and (2,2) respectively. Find the coordinates of the fourth vertex :

Options

A. a. (-1,2)
B. b. (-2,1)
C. c. (2,3)
D. d. (3,-2)

Answer

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

### Step-by-Step Solution:

1. **Identify the vertices**:
- Let A = (-1, 0)
- Let B = (3, 1)
- Let C = (2, 2)
- Let D = (x, y) be the fourth vertex we need to find.

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

3. **Find the midpoint of diagonal BD**:
The midpoint M of diagonal BD can be expressed as:
\[ M_{BD} = \left(\frac{3 + x}{2}, \frac{1 + y}{2}\right) \]

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

5. **Create equations from the midpoints**:
From the x-coordinates:
\[ \frac{3 + x}{2} = \frac{1}{2} \]
From the y-coordinates:
\[ \frac{1 + y}{2} = 1 \]

6. **Solve for x**:
Multiply both sides of the first equation by 2:
\[ 3 + x = 1 \]
Subtract 3 from both sides:
\[ x = 1 - 3 = -2 \]

7. **Solve for y**:
Multiply both sides of the second equation by 2:
\[ 1 + y = 2 \]
Subtract 1 from both sides:
\[ y = 2 - 1 = 1 \]

8. **Conclusion**:
The coordinates of the fourth vertex D are:
\[ D = (-2, 1) \]

### Final Answer:
The coordinates of the fourth vertex are (-2, 1).