What will be the perimeter of a rectangle if its length is 3 times its width and the length of the diagonal is `8sqrt(10)` cm?

Options

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

Answer

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

### Step 1: Define the Variables
Let the width (breadth) of the rectangle be \( x \). According to the problem, the length \( l \) is 3 times the width, so we can express the length as:
\[ l = 3x \]

### Step 2: Use the Diagonal Length
The length of the diagonal \( d \) is given as \( 8\sqrt{10} \) cm. According to the Pythagorean theorem, for a rectangle, the relationship between the length, width, and diagonal 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 \( (8\sqrt{10})^2 \):
\[ (8\sqrt{10})^2 = 64 \times 10 = 640 \]
Now substituting this back into the equation:
\[ 640 = (3x)^2 + x^2 \]
This simplifies to:
\[ 640 = 9x^2 + x^2 \]
\[ 640 = 10x^2 \]

### 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 gives:
\[ x = \sqrt{64} = 8 \text{ cm} \]

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

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

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

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