The length of a rectangle is thrice its breadth and the length of its diagonal is `8sqrt(10)` cm . The perimeter of the rectangle is

Options

A. `15sqrt(10)cm`
B. `16sqrt(10)cm`
C. `24 sqrt(10) cm `
D. 64 cm

Explanation

To find the perimeter of the rectangle, we will follow these steps:

### Step 1: Define the variables
Let the breadth of the rectangle be \( x \) cm. According to the problem, the length of the rectangle is thrice its breadth. Therefore, the length \( L \) can be expressed as:
\[ L = 3x \]

### Step 2: Use the diagonal length
We are given that the length of the diagonal of the rectangle is \( 8\sqrt{10} \) cm. According to the Pythagorean theorem, the relationship between the length, breadth, and diagonal \( D \) of a rectangle is given by:
\[ D^2 = L^2 + B^2 \]
Substituting the values we have:
\[ (8\sqrt{10})^2 = (3x)^2 + x^2 \]

### Step 3: Simplify the equation
Calculating the left side:
\[ (8\sqrt{10})^2 = 64 \times 10 = 640 \]
Now substituting this into the equation:
\[ 640 = (3x)^2 + x^2 \]
\[ 640 = 9x^2 + x^2 \]
\[ 640 = 10x^2 \]

### Step 4: Solve for \( x^2 \)
Now, we can solve for \( x^2 \):
\[ x^2 = \frac{640}{10} = 64 \]

### Step 5: Find \( x \)
Taking the square root of both sides:
\[ x = \sqrt{64} = 8 \text{ cm} \]

### Step 6: Calculate the length \( L \)
Now, we can find the length \( L \):
\[ L = 3x = 3 \times 8 = 24 \text{ cm} \]

### Step 7: Calculate the perimeter \( P \)
The formula for the perimeter \( P \) of a rectangle is given by:
\[ P = 2(L + B) \]
Substituting the values of \( L \) and \( B \):
\[ P = 2(24 + 8) = 2 \times 32 = 64 \text{ cm} \]

### Final Answer
The perimeter of the rectangle is \( 64 \) cm.

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