The co-ordinates of the vertices of a side of square are (4, -3) and (-1,-5). Its area is

Options

A. `2sqrt(29)` sq. units
B. `(sqrt(89))/(2)` sq. units
C. 89 sq. units
D. 29 sq. units

Answer

To find the area of the square given the coordinates of two of its vertices, we can follow these steps:

### Step 1: Identify the Coordinates
The coordinates of the two vertices of the square are given as:
- Vertex A: (4, -3)
- Vertex B: (-1, -5)

### Step 2: Use the Distance Formula
To find the length of the side of the square, we will use the distance formula, which is given by:
\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, we can assign:
- \( x_1 = 4 \), \( y_1 = -3 \)
- \( x_2 = -1 \), \( y_2 = -5 \)

### Step 3: Substitute the Values into the Formula
Now, substituting the values into the distance formula:
\[ AB = \sqrt{((-1) - 4)^2 + ((-5) - (-3))^2} \]
This simplifies to:
\[ AB = \sqrt{(-5)^2 + (-2)^2} \]

### Step 4: Calculate the Squares
Calculating the squares:
\[ AB = \sqrt{25 + 4} \]

### Step 5: Add the Results
Now, add the results:
\[ AB = \sqrt{29} \]

### Step 6: Calculate the Area of the Square
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Since the length of one side (AB) is \(\sqrt{29}\), the area will be:
\[ \text{Area} = (\sqrt{29})^2 = 29 \]

### Final Answer
Thus, the area of the square is:
\[ \text{Area} = 29 \text{ square units} \]

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