The length and breadth of a rectangle are in the ratio 8:3. Find Its area if perimeter is 132 cm.

Options

A. 864 `cm^2`
B. 684`cm^2`
C. 648 `cm^2`
D. 846 `cm^2`

Answer

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

### Step 1: Understand the Ratio
The length and breadth of the rectangle are given in the ratio 8:3. This means we can express the length (L) and breadth (B) in terms of a variable \( x \):
- Length \( L = 8x \)
- Breadth \( B = 3x \)

**Hint:** Ratios can be converted into equations by introducing a variable.

### Step 2: Use the Perimeter Formula
The perimeter (P) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
We know the perimeter is 132 cm. So we can set up the equation:
\[ 2(8x + 3x) = 132 \]

**Hint:** The perimeter formula combines both dimensions of the rectangle.

### Step 3: Simplify the Equation
Now, simplify the equation:
\[ 2(11x) = 132 \]
This simplifies to:
\[ 22x = 132 \]

**Hint:** Combine like terms to simplify the equation.

### Step 4: Solve for \( x \)
Now, divide both sides by 22 to find \( x \):
\[ x = \frac{132}{22} \]
\[ x = 6 \]

**Hint:** Isolate the variable by performing inverse operations.

### Step 5: Find Length and Breadth
Now that we have \( x \), we can find the length and breadth:
- Length \( L = 8x = 8 \times 6 = 48 \) cm
- Breadth \( B = 3x = 3 \times 6 = 18 \) cm

**Hint:** Substitute the value of \( x \) back into the expressions for length and breadth.

### Step 6: Calculate the Area
The area (A) of a rectangle is given by the formula:
\[ A = L \times B \]
Substituting the values we found:
\[ A = 48 \times 18 \]

Now, calculate the area:
\[ A = 864 \text{ cm}^2 \]

**Hint:** The area is found by multiplying the length and breadth.

### Final Answer
The area of the rectangle is \( 864 \text{ cm}^2 \).