The points
(i) `A(0,-1),B(2,1),C(0,3),D(-2,1)`
are the vertices of a

Options

A. square
B. rectangle
C. parallelogram
D. none

Answer

To determine the type of quadrilateral formed by the points A(0, -1), B(2, 1), C(0, 3), and D(-2, 1), we will follow these steps:

### Step 1: Find the slopes of the sides

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

**Calculating the slopes:**

1. **Slope of AB:**
\[ m_{AB} = \frac{1 - (-1)}{2 - 0} = \frac{2}{2} = 1 \]

2. **Slope of BC:**
\[ m_{BC} = \frac{3 - 1}{0 - 2} = \frac{2}{-2} = -1 \]

3. **Slope of CD:**
\[ m_{CD} = \frac{1 - 3}{-2 - 0} = \frac{-2}{-2} = 1 \]

4. **Slope of DA:**
\[ m_{DA} = \frac{-1 - 1}{0 - (-2)} = \frac{-2}{2} = -1 \]

### Step 2: Check for perpendicularity

Two lines are perpendicular if the product of their slopes is -1.

- **Check AB and BC:**
\[ m_{AB} \cdot m_{BC} = 1 \cdot (-1) = -1 \quad \text{(Perpendicular)} \]

- **Check CD and DA:**
\[ m_{CD} \cdot m_{DA} = 1 \cdot (-1) = -1 \quad \text{(Perpendicular)} \]

### Step 3: Check for equal lengths

To determine if the quadrilateral is a rectangle or square, we need to check the lengths of the sides.

**Length of AB:**
\[ AB = \sqrt{(2 - 0)^2 + (1 - (-1))^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

**Length of BC:**
\[ BC = \sqrt{(0 - 2)^2 + (3 - 1)^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

**Length of CD:**
\[ CD = \sqrt{(-2 - 0)^2 + (1 - 3)^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

**Length of DA:**
\[ DA = \sqrt{(0 - (-2))^2 + (-1 - 1)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

### Step 4: Conclusion

Since:
- All sides are equal (AB = BC = CD = DA = \( 2\sqrt{2} \))
- Opposite sides are parallel and the angles between adjacent sides are 90 degrees

The quadrilateral ABCD is a **square**.