A(-2,-1) , B(1,0) , C(4,3) and D(1,2) are the four points of a quadrilateral . The quadrilateral is a :

Options

A. Square
B. Rhombus
C. Parallelogram
D. none of (a) , (b) ,(c)

Answer

To determine the type of quadrilateral formed by the points A(-2,-1), B(1,0), C(4,3), and D(1,2), we will follow these steps:

### Step 1: Plot the Points
We will plot the points A, B, C, and D on the Cartesian plane.

- A(-2, -1): This point is located 2 units left and 1 unit down from the origin.
- B(1, 0): This point is located 1 unit right from the origin.
- C(4, 3): This point is located 4 units right and 3 units up from the origin.
- D(1, 2): This point is located 1 unit right and 2 units up from the origin.

### Step 2: Connect the Points
Next, we will connect the points in the order A, B, C, D, and back to A to form the quadrilateral.

### Step 3: Calculate the Lengths of the Sides
We will calculate the lengths of the sides AB, BC, CD, and DA using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

1. **Length of AB**:
\[ AB = \sqrt{(1 - (-2))^2 + (0 - (-1))^2} = \sqrt{(1 + 2)^2 + (0 + 1)^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \]

2. **Length of BC**:
\[ BC = \sqrt{(4 - 1)^2 + (3 - 0)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

3. **Length of CD**:
\[ CD = \sqrt{(1 - 4)^2 + (2 - 3)^2} = \sqrt{(-3)^2 + (-1)^2} = \sqrt{9 + 1} = \sqrt{10} \]

4. **Length of DA**:
\[ DA = \sqrt{(-2 - 1)^2 + (-1 - 2)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

### Step 4: Check for Parallel Sides
To determine if the quadrilateral is a parallelogram, we need to check if opposite sides are equal:
- AB = CD = \(\sqrt{10}\)
- BC = DA = \(3\sqrt{2}\)

Since opposite sides are equal, the quadrilateral is a parallelogram.

### Step 5: Check for Rhombus
To check if it is a rhombus, we need to see if all sides are equal. Since AB and CD are equal, and BC and DA are equal but not equal to AB and CD, it is not a rhombus.

### Conclusion
The quadrilateral formed by points A, B, C, and D is a parallelogram.