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 two things:

1. The quadrilateral formed by these points is a parallelogram.
2. The angles between adjacent sides are 90 degrees.

### Step 1: Prove that the quadrilateral is a parallelogram

A quadrilateral is a parallelogram if the midpoints of its diagonals are the same. We will find the midpoints of diagonals AC and BD.

**Finding the midpoint of AC:**
- A = (-1, -2)
- C = (2, 5)

Midpoint of AC = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
= \(\left(\frac{-1 + 2}{2}, \frac{-2 + 5}{2}\right)\)
= \(\left(\frac{1}{2}, \frac{3}{2}\right)\)

**Finding the midpoint of BD:**
- B = (4, 3)
- D = (-3, 0)

Midpoint of BD = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\)
= \(\left(\frac{4 - 3}{2}, \frac{3 + 0}{2}\right)\)
= \(\left(\frac{1}{2}, \frac{3}{2}\right)\)

Since the midpoints of AC and BD are the same, we conclude that the quadrilateral ABCD is a parallelogram.

### Step 2: Prove that the angles between adjacent sides are 90 degrees

To show that the angle between adjacent sides is 90 degrees, we can use the distance formula to find the lengths of the sides and then apply the Pythagorean theorem.

**Finding the lengths of the sides:**

1. **Length of DC:**
- D = (-3, 0)
- C = (2, 5)

Using the distance formula:
\[ DC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(2 - (-3))^2 + (5 - 0)^2} = \sqrt{(2 + 3)^2 + 5^2} = \sqrt{5^2 + 5^2} = \sqrt{50} \]

2. **Length of BC:**
- B = (4, 3)
- C = (2, 5)

Using the distance formula:
\[ BC = \sqrt{(2 - 4)^2 + (5 - 3)^2} = \sqrt{(-2)^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \]

3. **Length of BD:**
- B = (4, 3)
- D = (-3, 0)

Using the distance formula:
\[ BD = \sqrt{(-3 - 4)^2 + (0 - 3)^2} = \sqrt{(-7)^2 + (-3)^2} = \sqrt{49 + 9} = \sqrt{58} \]

### Step 3: Verify using Pythagorean theorem

According to the Pythagorean theorem, if ABCD is a rectangle, then:
\[ DC^2 + BC^2 = BD^2 \]

Calculating:
- \(DC^2 = 50\)
- \(BC^2 = 8\)
- \(BD^2 = 58\)

Now check:
\[ 50 + 8 = 58 \]

Since this holds true, we conclude that angle C is 90 degrees.

### Conclusion
Since we have shown that ABCD is a parallelogram and that the angles between adjacent sides are 90 degrees, we can conclude that the points A (-1,-2), B (4,3), C (2,5), and D (-3,0) form a rectangle.

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