The quadrilateral formed by the points (-1, -2), (1,0), (-1, 2) and (-3,0) is a

Options

A. rectangle
B. square
C. rhombus
D. none of these

Answer

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

### Step 1: Identify the Points
Let:
- A = (-1, -2)
- B = (1, 0)
- C = (-1, 2)
- D = (-3, 0)

### Step 2: Calculate the Length of Each Side
We will use the distance formula to find the lengths of the sides AB, BC, CD, and DA. The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

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

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

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

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

### Step 3: Check the Lengths of the Sides
We found that:
- AB = 2√2
- BC = 2√2
- CD = 2√2
- DA = 2√2

Since all four sides are equal, we have established that the quadrilateral is either a square or a rhombus.

### Step 4: Calculate the Lengths of the Diagonals
Next, we will calculate the lengths of the diagonals AC and BD.

**Calculate AC:**
\[ AC = \sqrt{(-1 - (-1))^2 + (2 - (-2))^2} = \sqrt{(0)^2 + (4)^2} = \sqrt{16} = 4 \]

**Calculate BD:**
\[ BD = \sqrt{(1 - (-3))^2 + (0 - 0)^2} = \sqrt{(4)^2 + (0)^2} = \sqrt{16} = 4 \]

### Step 5: Check the Lengths of the Diagonals
We found that:
- AC = 4
- BD = 4

Since both diagonals are equal, we conclude that the quadrilateral is a square.

### Final Conclusion
The quadrilateral formed by the points (-1, -2), (1, 0), (-1, 2), and (-3, 0) is a square.

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