If `A(2,-1),B(3,4),C(-2,3)andD(-3,-2)` be four points in a plane show tht ABCD is a rhombus but not a square. Find the area of the rhombus.

Answer

To determine whether the points A(2, -1), B(3, 4), C(-2, 3), and D(-3, -2) form a rhombus but not a square, and to find the area of the rhombus, we will follow these steps:

### Step 1: Calculate the lengths of the sides

We will use the distance formula to find the lengths of the sides AB, BC, CD, and DA.

The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

1. **Length of AB**:
\[ AB = \sqrt{(3 - 2)^2 + (4 - (-1))^2} = \sqrt{(1)^2 + (5)^2} = \sqrt{1 + 25} = \sqrt{26} \]

2. **Length of BC**:
\[ BC = \sqrt{(-2 - 3)^2 + (3 - 4)^2} = \sqrt{(-5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

3. **Length of CD**:
\[ CD = \sqrt{(-3 - (-2))^2 + (-2 - 3)^2} = \sqrt{(-1)^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26} \]

4. **Length of DA**:
\[ DA = \sqrt{(2 - (-3))^2 + (-1 - (-2))^2} = \sqrt{(5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

### Step 2: Verify that all sides are equal

From the calculations, we find:
- \( AB = BC = CD = DA = \sqrt{26} \)

Since all four sides are equal, ABCD is a rhombus.

### Step 3: Calculate the lengths of the diagonals

Next, we will calculate the lengths of the diagonals AC and BD.

1. **Length of AC**:
\[ AC = \sqrt{(-2 - 2)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} \]

2. **Length of BD**:
\[ BD = \sqrt{(-3 - 3)^2 + (-2 - 4)^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} \]

### Step 4: Verify that the diagonals are not equal

From the calculations, we find:
- \( AC = \sqrt{32} \)
- \( BD = \sqrt{72} \)

Since the diagonals are not equal, ABCD is not a square.

### Step 5: Calculate the area of the rhombus

The area \( A \) of a rhombus can be calculated using the formula:
\[ A = \frac{1}{2} \times d_1 \times d_2 \]
where \( d_1 \) and \( d_2 \) are the lengths of the diagonals.

Substituting the lengths of the diagonals:
\[ A = \frac{1}{2} \times \sqrt{32} \times \sqrt{72} \]

Calculating the product:
\[ \sqrt{32} \times \sqrt{72} = \sqrt{32 \times 72} = \sqrt{2304} \]

Now, we find the square root:
\[ \sqrt{2304} = 48 \]

Thus, the area is:
\[ A = \frac{1}{2} \times 48 = 24 \]

### Final Answer
The area of the rhombus ABCD is **24 square units**.

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