The points `(-2, -1), (1, 0), (4, 3)` and `(1, 2)` are

Options

A. collinear
B. concyclic
C. the vertices of a parallelogram
D. the vertices of a rectangle

Answer

To determine the relationship between the points \((-2, -1)\), \((1, 0)\), \((4, 3)\), and \((1, 2)\), we will check if they form a parallelogram or a rectangle.

### Step 1: Calculate the midpoints of the diagonals
First, we will find the midpoints of the diagonals formed by the pairs of points.

1. **Midpoint of points \((-2, -1)\) and \((4, 3)\)**:
\[ \text{Midpoint} = \left(\frac{-2 + 4}{2}, \frac{-1 + 3}{2}\right) = \left(\frac{2}{2}, \frac{2}{2}\right) = (1, 1) \]

2. **Midpoint of points \((1, 0)\) and \((1, 2)\)**:
\[ \text{Midpoint} = \left(\frac{1 + 1}{2}, \frac{0 + 2}{2}\right) = \left(\frac{2}{2}, \frac{2}{2}\right) = (1, 1) \]

### Step 2: Compare the midpoints
Since both midpoints are equal, we conclude that the points form a parallelogram.

### Step 3: Check if it is a rectangle
To determine if the parallelogram is a rectangle, we need to check if the lengths of the diagonals are equal.

1. **Calculate the distance between points \((-2, -1)\) and \((4, 3)\)**:
\[ d_1 = \sqrt{(4 - (-2))^2 + (3 - (-1))^2} = \sqrt{(4 + 2)^2 + (3 + 1)^2} = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} \]

2. **Calculate the distance between points \((1, 0)\) and \((1, 2)\)**:
\[ d_2 = \sqrt{(1 - 1)^2 + (2 - 0)^2} = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \]

### Step 4: Compare the lengths of the diagonals
Since \(d_1 = \sqrt{52}\) and \(d_2 = 2\), the lengths of the diagonals are not equal.

### Conclusion
The points \((-2, -1)\), \((1, 0)\), \((4, 3)\), and \((1, 2)\) are the vertices of a parallelogram, but they do not form a rectangle.

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