If three vertices of a parallelogram taken in order are `(-1, 0), (3, 2) and (2, 3)` then area of the parallelogram is

Options

A. `12`
B. `(21)/(2)`
C. `6`
D. `3`

Explanation

To find the area of the parallelogram given three vertices, we can follow these steps:

### Step 1: Identify the vertices
Let the vertices of the parallelogram be:
- A = (-1, 0)
- B = (3, 2)
- C = (2, 3)

We need to find the coordinates of the fourth vertex D.

### Step 2: Find the coordinates of the fourth vertex D
The diagonals of a parallelogram bisect each other. Therefore, we can find the midpoint of the diagonal AC and set it equal to the midpoint of the diagonal BD.

1. **Calculate the midpoint of AC:**
\[ \text{Midpoint of AC} = \left( \frac{-1 + 2}{2}, \frac{0 + 3}{2} \right) = \left( \frac{1}{2}, \frac{3}{2} \right) \]

2. **Let the coordinates of D be (x, y). Calculate the midpoint of BD:**
\[ \text{Midpoint of BD} = \left( \frac{3 + x}{2}, \frac{2 + y}{2} \right) \]

3. **Set the midpoints equal to each other:**
\[ \frac{3 + x}{2} = \frac{1}{2} \quad \text{and} \quad \frac{2 + y}{2} = \frac{3}{2} \]

4. **Solve for x:**
\[ 3 + x = 1 \implies x = 1 - 3 = -2 \]

5. **Solve for y:**
\[ 2 + y = 3 \implies y = 3 - 2 = 1 \]

Thus, the coordinates of D are (-2, 1).

### Step 3: Calculate the area of the parallelogram
The area of a parallelogram can be calculated using the formula:
\[ \text{Area} = \left| \vec{AB} \times \vec{AC} \right| \]
where \(\vec{AB}\) and \(\vec{AC}\) are the vectors formed by the vertices.

1. **Calculate vectors AB and AC:**
\[ \vec{AB} = B - A = (3 - (-1), 2 - 0) = (4, 2) \]
\[ \vec{AC} = C - A = (2 - (-1), 3 - 0) = (3, 3) \]

2. **Calculate the cross product \(\vec{AB} \times \vec{AC}\):**
The area can be calculated using the determinant:
\[ \text{Area} = \left| \begin{vmatrix} 4 & 2 \\ 3 & 3 \end{vmatrix} \right| = |(4 \cdot 3) - (2 \cdot 3)| = |12 - 6| = |6| = 6 \]

### Final Answer
The area of the parallelogram is **6 square units**.

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