The length of a rectangular plot is twice of its width. If the length of a diagonal is `9sqrt5` metres, the perimeter of the rectangle is -----

Options

A. A)27 m
B. B)54 m
C. C)81 m
D. D)None of these

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the Variables
Let the width of the rectangle be \( K \) meters. Since the length is twice the width, we can express the length as:
\[ \text{Length} = 2K \text{ meters} \]

**Hint:** Start by defining the variables based on the relationships given in the problem.

### Step 2: Use the Pythagorean Theorem
According to the Pythagorean theorem, the diagonal \( D \) of a rectangle can be calculated using the formula:
\[ D = \sqrt{(\text{Length})^2 + (\text{Width})^2} \]
Given that the diagonal is \( 9\sqrt{5} \) meters, we can set up the equation:
\[ 9\sqrt{5} = \sqrt{(2K)^2 + K^2} \]

**Hint:** Remember to apply the Pythagorean theorem to relate the sides of the rectangle to the diagonal.

### Step 3: Square Both Sides
To eliminate the square root, we square both sides of the equation:
\[ (9\sqrt{5})^2 = (2K)^2 + K^2 \]
This simplifies to:
\[ 81 \cdot 5 = 4K^2 + K^2 \]
\[ 405 = 5K^2 \]

**Hint:** Squaring both sides helps to simplify the equation by removing the square root.

### Step 4: Solve for \( K^2 \)
Now, we can solve for \( K^2 \):
\[ K^2 = \frac{405}{5} = 81 \]
Taking the square root gives:
\[ K = \sqrt{81} = 9 \text{ meters} \]

**Hint:** Isolate \( K^2 \) and then take the square root to find \( K \).

### Step 5: Find the Length
Now that we have the width, we can find the length:
\[ \text{Length} = 2K = 2 \times 9 = 18 \text{ meters} \]

**Hint:** Use the relationship between length and width to find the length.

### Step 6: Calculate the Perimeter
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2 \times (\text{Length} + \text{Width}) \]
Substituting the values we found:
\[ P = 2 \times (18 + 9) = 2 \times 27 = 54 \text{ meters} \]

**Hint:** Use the perimeter formula and substitute the length and width to find the perimeter.

### Final Answer
The perimeter of the rectangle is \( 54 \) meters.