The length of a rectangular garden is 12 metres and its breadth is 5 metres. Find tne length of the diagonal of a square garden having the same area as that of the rectangular garden

Options

A. `2 sqrt(30)` m
B. `sqrt(3) m`
C. `13 m`
D. `8 sqrt(15)m`

Answer

To solve the problem step by step, we will find the area of the rectangular garden, then determine the side length of a square garden with the same area, and finally calculate the length of the diagonal of that square garden.

### Step 1: Calculate the area of the rectangular garden.
The area \( A \) of a rectangle is given by the formula:
\[ A = \text{length} \times \text{breadth} \]
Given that the length is 12 meters and the breadth is 5 meters, we can substitute these values into the formula:
\[ A = 12 \, \text{m} \times 5 \, \text{m} = 60 \, \text{m}^2 \]

### Step 2: Determine the side length of the square garden.
Since the square garden has the same area as the rectangular garden, we set the area of the square equal to the area of the rectangle:
\[ \text{Area of square} = \text{side}^2 = 60 \, \text{m}^2 \]
To find the side length of the square, we take the square root of the area:
\[ \text{side} = \sqrt{60} \, \text{m} \]

### Step 3: Calculate the diagonal of the square garden.
The diagonal \( d \) of a square can be calculated using the formula:
\[ d = \text{side} \times \sqrt{2} \]
Substituting the value of the side we found:
\[ d = \sqrt{60} \times \sqrt{2} = \sqrt{60 \times 2} = \sqrt{120} \]
We can simplify \( \sqrt{120} \):
\[ \sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30} \]

### Final Answer:
The length of the diagonal of the square garden is:
\[ 2\sqrt{30} \, \text{meters} \]