Area of a rectangle is `180cm^(2)` and difference between length and breadth of rectangle is 3cm. If breadth of rectangle is equal to the side of a square, then find perimeter of square?

Options

A. A)44 cm
B. B)52 cm
C. C)48 cm
D. D)64 cm

Answer

To solve the problem step by step, we will use the information given about the rectangle and the relationship to the square.

### Step 1: Understand the given information
- The area of the rectangle is \(180 \, \text{cm}^2\).
- The difference between the length (L) and the breadth (B) of the rectangle is \(3 \, \text{cm}\).
- The breadth of the rectangle is equal to the side of a square.

### Step 2: Set up the equations
From the information given, we can set up the following equations:
1. \(L \times B = 180 \, \text{cm}^2\) (Area of the rectangle)
2. \(L - B = 3 \, \text{cm}\) (Difference between length and breadth)

### Step 3: Express L in terms of B
From the second equation, we can express the length in terms of the breadth:
\[ L = B + 3 \]

### Step 4: Substitute L in the area equation
Now, substitute \(L\) in the area equation:
\[ (B + 3) \times B = 180 \]
Expanding this gives:
\[ B^2 + 3B = 180 \]

### Step 5: Rearrange the equation
Rearranging the equation gives us a standard quadratic equation:
\[ B^2 + 3B - 180 = 0 \]

### Step 6: Factor the quadratic equation
Next, we need to factor the quadratic equation. We are looking for two numbers that multiply to \(-180\) and add to \(3\). The factors are \(15\) and \(-12\):
\[ (B + 15)(B - 12) = 0 \]

### Step 7: Solve for B
Setting each factor to zero gives us:
1. \(B + 15 = 0 \Rightarrow B = -15\) (not valid since breadth cannot be negative)
2. \(B - 12 = 0 \Rightarrow B = 12\)

Thus, the breadth \(B\) of the rectangle is \(12 \, \text{cm}\).

### Step 8: Find L using B
Now, substitute \(B\) back to find \(L\):
\[ L = B + 3 = 12 + 3 = 15 \, \text{cm} \]

### Step 9: Find the side of the square
Since the breadth of the rectangle is equal to the side of the square, the side of the square is:
\[ \text{Side of square} = 12 \, \text{cm} \]

### Step 10: Calculate the perimeter of the square
The perimeter \(P\) of a square is given by:
\[ P = 4 \times \text{Side} \]
Substituting the side length:
\[ P = 4 \times 12 = 48 \, \text{cm} \]

### Final Answer
The perimeter of the square is \(48 \, \text{cm}\).

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