If P(2, -1), Q(3, 4), R(-2, 3) and S(-3, -2) are four points in a plane, show that PQRS is a rhombus but not a square. Also find its area.

Answer

To show that the quadrilateral PQRS is a rhombus but not a square, and to find its area, we will follow these steps:

### Step 1: Find the lengths of all sides
We will use the distance formula to calculate the lengths of the sides PQ, QR, RS, and SP.

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

**Coordinates:**
- \( P(2, -1) \)
- \( Q(3, 4) \)
- \( R(-2, 3) \)
- \( S(-3, -2) \)

**Calculating PQ:**
\[ PQ = \sqrt{(3 - 2)^2 + (4 - (-1))^2} = \sqrt{(1)^2 + (5)^2} = \sqrt{1 + 25} = \sqrt{26} \]

**Calculating QR:**
\[ QR = \sqrt{(-2 - 3)^2 + (3 - 4)^2} = \sqrt{(-5)^2 + (-1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

**Calculating RS:**
\[ RS = \sqrt{(-3 - (-2))^2 + (-2 - 3)^2} = \sqrt{(-1)^2 + (-5)^2} = \sqrt{1 + 25} = \sqrt{26} \]

**Calculating SP:**
\[ SP = \sqrt{(2 - (-3))^2 + (-1 - (-2))^2} = \sqrt{(5)^2 + (1)^2} = \sqrt{25 + 1} = \sqrt{26} \]

### Conclusion for Step 1:
All sides are equal:
\[ PQ = QR = RS = SP = \sqrt{26} \]

### Step 2: Find the lengths of the diagonals
Next, we will find the lengths of the diagonals PR and QS.

**Calculating PR:**
\[ PR = \sqrt{(-2 - 2)^2 + (3 - (-1))^2} = \sqrt{(-4)^2 + (4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]

**Calculating QS:**
\[ QS = \sqrt{(-3 - 3)^2 + (-2 - 4)^2} = \sqrt{(-6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2} \]

### Conclusion for Step 2:
The diagonals are:
\[ PR = 4\sqrt{2}, \quad QS = 6\sqrt{2} \]
Since the diagonals are not equal, PQRS is not a square.

### Step 3: Prove that PQRS is a rhombus
Since all sides are equal and the diagonals are not equal, we conclude that PQRS is a rhombus.

### Step 4: Find 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 values:
\[ A = \frac{1}{2} \times (4\sqrt{2}) \times (6\sqrt{2}) = \frac{1}{2} \times 24 = 12 \text{ square units} \]

### Final Conclusion:
The area of the rhombus PQRS is \(12\) square units.

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