Two square have sides x cma nd (2 x + 1) cm, respectively . The sum of their perimeters is 100 cm. Area (in `cm^(2)` ) of the bigger square is

Options

A. 225
B. 289
C. 64
D. 81

Answer

To solve the problem, we need to find the area of the bigger square given the sides of two squares and the sum of their perimeters. Here’s a step-by-step solution:

### Step 1: Define the sides of the squares
Let the side of the first square be \( x \) cm.
Let the side of the second square be \( (2x + 1) \) cm.

### Step 2: Write the formula for the perimeter of each square
The perimeter of a square is given by the formula \( P = 4 \times \text{side} \).
Thus, the perimeter of the first square is \( 4x \) cm, and the perimeter of the second square is \( 4(2x + 1) \) cm.

### Step 3: Set up the equation for the sum of the perimeters
According to the problem, the sum of the perimeters of the two squares is 100 cm.
So, we can write the equation:
\[ 4x + 4(2x + 1) = 100 \]

### Step 4: Simplify the equation
Expanding the equation gives:
\[ 4x + 8x + 4 = 100 \]
Combining like terms results in:
\[ 12x + 4 = 100 \]

### Step 5: Solve for \( x \)
Subtract 4 from both sides:
\[ 12x = 96 \]
Now, divide both sides by 12:
\[ x = 8 \text{ cm} \]

### Step 6: Find the side of the bigger square
Now that we have \( x \), we can find the side of the bigger square:
\[ \text{Side of the bigger square} = 2x + 1 = 2(8) + 1 = 16 + 1 = 17 \text{ cm} \]

### Step 7: Calculate the area of the bigger square
The area of a square is given by the formula \( \text{Area} = \text{side}^2 \).
Thus, the area of the bigger square is:
\[ \text{Area} = 17 \times 17 = 289 \text{ cm}^2 \]

### Final Answer
The area of the bigger square is \( 289 \text{ cm}^2 \).

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