Points (4, -1), (6, 0), (7, 2) and (5, 1) are joined to be a vertex of a quadrilateral. What will be the structure?

Options

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

Answer

To determine the structure of the quadrilateral formed by the points (4, -1), (6, 0), (7, 2), and (5, 1), we will follow these steps:

### Step 1: Identify the Points
Let the points be:
- A(4, -1)
- B(6, 0)
- C(7, 2)
- D(5, 1)

### Step 2: Calculate the Length of Each Side
We will use the distance formula to calculate the lengths of the sides of the quadrilateral. The distance formula between two points (x1, y1) and (x2, y2) is given by:

\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

#### Length of AB
\[ AB = \sqrt{(6 - 4)^2 + (0 - (-1))^2} = \sqrt{(2)^2 + (1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

#### Length of BC
\[ BC = \sqrt{(7 - 6)^2 + (2 - 0)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

#### Length of CD
\[ CD = \sqrt{(5 - 7)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

#### Length of DA
\[ DA = \sqrt{(4 - 5)^2 + (-1 - 1)^2} = \sqrt{(-1)^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 3: Analyze the Lengths
We find that:
- AB = BC = CD = DA = \(\sqrt{5}\)

Since all sides are equal, the quadrilateral could be a rhombus or a square.

### Step 4: Check the Diagonals
To determine if it is a rhombus or a square, we need to check the diagonals.

#### Diagonal AC
\[ AC = \sqrt{(7 - 4)^2 + (2 - (-1))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \]

#### Diagonal BD
\[ BD = \sqrt{(5 - 6)^2 + (1 - 0)^2} = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

### Step 5: Determine the Type of Quadrilateral
Since the diagonals are not equal (AC ≠ BD), the quadrilateral cannot be a square. However, since all sides are equal, it is a rhombus.

### Conclusion
The structure formed by the points (4, -1), (6, 0), (7, 2), and (5, 1) is a **rhombus**.

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