The length of a rectangle is three times its width and the length of its diagonal is `6sqrt(10)` cm. The perimeter of the rectangle is

Options

A. `48 cm`
B. `36 cm`
C. `24 cm`
D. `24sqrt(10)cm`

Answer

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

### Step 1: Define the Variables
Let the width of the rectangle be \( x \) cm. According to the problem, the length of the rectangle is three times its width. Therefore, the length \( l \) can be expressed as:
\[ l = 3x \]

### Step 2: Use the Diagonal Information
The diagonal \( d \) of the rectangle is given as \( 6\sqrt{10} \) cm. According to the Pythagorean theorem, for a rectangle, the relationship between the length, width, and diagonal is:
\[ d^2 = l^2 + w^2 \]
Substituting the values we have:
\[ (6\sqrt{10})^2 = (3x)^2 + x^2 \]

### Step 3: Calculate the Square of the Diagonal
Now, calculate \( (6\sqrt{10})^2 \):
\[ (6\sqrt{10})^2 = 36 \times 10 = 360 \]

### Step 4: Substitute and Simplify
Now substitute back into the equation:
\[ 360 = (3x)^2 + x^2 \]
This expands to:
\[ 360 = 9x^2 + x^2 \]
Combining the terms gives:
\[ 360 = 10x^2 \]

### Step 5: Solve for \( x^2 \)
To find \( x^2 \), divide both sides by 10:
\[ x^2 = \frac{360}{10} = 36 \]

### Step 6: Find \( x \)
Taking the square root of both sides gives:
\[ x = \sqrt{36} = 6 \text{ cm} \]

### Step 7: Find the Length
Now, substitute \( x \) back to find the length:
\[ l = 3x = 3 \times 6 = 18 \text{ cm} \]

### Step 8: Calculate the Perimeter
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(l + w) \]
Substituting the values of length and width:
\[ P = 2(18 + 6) = 2 \times 24 = 48 \text{ cm} \]

### Final Answer
Thus, the perimeter of the rectangle is:
\[ \boxed{48 \text{ cm}} \]

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