The points A (- 4, - 1), B (-2, - 4), C (4, 0) and D (2, 3) are the vertices of a

Options

A. Parallelogram
B. Rectangle
C. Rhombus
D. Square

Answer

To determine the type of quadrilateral formed by the points A (-4, -1), B (-2, -4), C (4, 0), and D (2, 3), we will calculate the lengths of the sides and the diagonals. We will then analyze these lengths to identify the properties of the quadrilateral.

### Step-by-Step Solution:

1. **Calculate the Length of AB:**
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-2 - (-4))^2 + (-4 - (-1))^2} \]
\[ = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

2. **Calculate the Length of BC:**
\[ BC = \sqrt{(4 - (-2))^2 + (0 - (-4))^2} = \sqrt{(6)^2 + (4)^2} \]
\[ = \sqrt{36 + 16} = \sqrt{52} \]

3. **Calculate the Length of CD:**
\[ CD = \sqrt{(2 - 4)^2 + (3 - 0)^2} = \sqrt{(-2)^2 + (3)^2} \]
\[ = \sqrt{4 + 9} = \sqrt{13} \]

4. **Calculate the Length of DA:**
\[ DA = \sqrt{(-4 - 2)^2 + (-1 - 3)^2} = \sqrt{(-6)^2 + (-4)^2} \]
\[ = \sqrt{36 + 16} = \sqrt{52} \]

5. **Calculate the Length of Diagonal AC:**
\[ AC = \sqrt{(4 - (-4))^2 + (0 - (-1))^2} = \sqrt{(8)^2 + (1)^2} \]
\[ = \sqrt{64 + 1} = \sqrt{65} \]

6. **Calculate the Length of Diagonal BD:**
\[ BD = \sqrt{(2 - (-2))^2 + (3 - (-4))^2} = \sqrt{(4)^2 + (7)^2} \]
\[ = \sqrt{16 + 49} = \sqrt{65} \]

7. **Analyze the Lengths:**
- We found that:
- \( AB = CD = \sqrt{13} \)
- \( BC = DA = \sqrt{52} \)
- \( AC = BD = \sqrt{65} \)

8. **Conclusion:**
Since both pairs of opposite sides are equal and the diagonals are also equal, the quadrilateral formed by points A, B, C, and D is a rectangle.

### Final Answer:
The quadrilateral formed by the points A (-4, -1), B (-2, -4), C (4, 0), and D (2, 3) is a rectangle.