The points A (-1,-2), B (4,3) ,C (2,5) and D (-3,0) in that order form a rectangle.

Explanation

To prove that the points A (-1, -2), B (4, 3), C (2, 5), and D (-3, 0) form a rectangle, we need to show that opposite sides are equal and parallel, and that the angles between adjacent sides are 90 degrees.

### Step-by-Step Solution:

**Step 1: Calculate the distances between the points.**

1. **Distance AB:**
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - (-1))^2 + (3 - (-2))^2} = \sqrt{(4 + 1)^2 + (3 + 2)^2} = \sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]

2. **Distance BC:**
\[ BC = \sqrt{(2 - 4)^2 + (5 - 3)^2} = \sqrt{(-2)^2 + (2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

3. **Distance CD:**
\[ CD = \sqrt{(-3 - 2)^2 + (0 - 5)^2} = \sqrt{(-5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]

4. **Distance DA:**
\[ DA = \sqrt{(-1 - (-3))^2 + (-2 - 0)^2} = \sqrt{(2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

**Step 2: Check if opposite sides are equal.**

From the calculations:
- \( AB = CD = 5\sqrt{2} \)
- \( BC = DA = 2\sqrt{2} \)

Thus, opposite sides are equal.

**Step 3: Calculate the slopes to check for right angles.**

1. **Slope of AB:**
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-2)}{4 - (-1)} = \frac{5}{5} = 1 \]

2. **Slope of BC:**
\[ m_{BC} = \frac{5 - 3}{2 - 4} = \frac{2}{-2} = -1 \]

3. **Slope of CD:**
\[ m_{CD} = \frac{0 - 5}{-3 - 2} = \frac{-5}{-5} = 1 \]

4. **Slope of DA:**
\[ m_{DA} = \frac{-2 - 0}{-1 - (-3)} = \frac{-2}{2} = -1 \]

**Step 4: Check if the product of slopes of adjacent sides equals -1.**

- For sides AB and BC:
\[ m_{AB} \cdot m_{BC} = 1 \cdot (-1) = -1 \quad \text{(90 degrees)} \]

- For sides BC and CD:
\[ m_{BC} \cdot m_{CD} = -1 \cdot 1 = -1 \quad \text{(90 degrees)} \]

- For sides CD and DA:
\[ m_{CD} \cdot m_{DA} = 1 \cdot (-1) = -1 \quad \text{(90 degrees)} \]

- For sides DA and AB:
\[ m_{DA} \cdot m_{AB} = -1 \cdot 1 = -1 \quad \text{(90 degrees)} \]

Since the product of the slopes of each pair of adjacent sides is -1, all angles are 90 degrees.

### Conclusion:
Since opposite sides are equal and parallel, and all angles are 90 degrees, the points A, B, C, and D form a rectangle.

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