Find the area of a rectangle whose vertices are A(-2, 6), B (5,3), C (-1,-11) and D(-8,-8) 

Options

A. `4sqrt(29)` sq. units
B. 116 sq. units
C. `29sqrt(5)` sq. units
D. `58sqrt(2)` sq. units

Answer

To find the area of the rectangle with vertices A(-2, 6), B(5, 3), C(-1, -11), and D(-8, -8), we will follow these steps:

### Step 1: Identify the vertices
The vertices of the rectangle are given as:
- A(-2, 6)
- B(5, 3)
- C(-1, -11)
- D(-8, -8)

### Step 2: Determine the length and breadth
To find the area of the rectangle, we need to calculate the lengths of two adjacent sides (length and breadth). We can use the distance formula to find the lengths AB and AD.

### Step 3: Use the distance formula
The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

### Step 4: Calculate the length AB
Using the coordinates of points A and B:
- A(-2, 6) → (x1, y1) = (-2, 6)
- B(5, 3) → (x2, y2) = (5, 3)

Now, substituting into the distance formula:
\[ AB = \sqrt{(5 - (-2))^2 + (3 - 6)^2} \]
\[ = \sqrt{(5 + 2)^2 + (3 - 6)^2} \]
\[ = \sqrt{(7)^2 + (-3)^2} \]
\[ = \sqrt{49 + 9} \]
\[ = \sqrt{58} \]

### Step 5: Calculate the length AD
Using the coordinates of points A and D:
- D(-8, -8) → (x2, y2) = (-8, -8)

Now, substituting into the distance formula:
\[ AD = \sqrt{(-8 - (-2))^2 + (-8 - 6)^2} \]
\[ = \sqrt{(-8 + 2)^2 + (-8 - 6)^2} \]
\[ = \sqrt{(-6)^2 + (-14)^2} \]
\[ = \sqrt{36 + 196} \]
\[ = \sqrt{232} \]

### Step 6: Calculate the area of the rectangle
The area of the rectangle is given by:
\[ \text{Area} = \text{Length} \times \text{Breadth} = AB \times AD \]
Substituting the values we found:
\[ \text{Area} = \sqrt{58} \times \sqrt{232} \]
\[ = \sqrt{58 \times 232} \]

### Step 7: Calculate \(58 \times 232\)
Calculating \(58 \times 232\):
\[ 58 \times 232 = 13456 \]
Thus,
\[ \text{Area} = \sqrt{13456} \]

### Step 8: Simplify \(\sqrt{13456}\)
Finding the square root:
\[ \sqrt{13456} \approx 116 \]

### Final Answer
The area of the rectangle is approximately \(116\) square units.

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