A square contains four times the area of another square. If one side of the larger square be 4 cm greater than that of smaller square, then the perimeter of smaller square will be equal to

Options

A. 8 cm
B. 16 cm
C. 24 cm
D. 32 cm

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the variables
Let the side of the smaller square be \( x \) cm.

### Step 2: Express the side of the larger square
According to the problem, the side of the larger square is \( x + 4 \) cm (since it is 4 cm greater than the side of the smaller square).

### Step 3: Write the area equations
The area of the smaller square is given by:
\[ \text{Area of smaller square} = x^2 \]
The area of the larger square is given by:
\[ \text{Area of larger square} = (x + 4)^2 \]
According to the problem, the area of the larger square is four times the area of the smaller square:
\[ (x + 4)^2 = 4x^2 \]

### Step 4: Expand the equation
Now, we will expand the left side of the equation:
\[ (x + 4)^2 = x^2 + 8x + 16 \]
So we have:
\[ x^2 + 8x + 16 = 4x^2 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 0 = 4x^2 - x^2 - 8x - 16 \]
This simplifies to:
\[ 3x^2 - 8x - 16 = 0 \]

### Step 6: Solve the quadratic equation
Now we will solve the quadratic equation \( 3x^2 - 8x - 16 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 3 \), \( b = -8 \), and \( c = -16 \):
\[ x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 3 \cdot (-16)}}{2 \cdot 3} \]
Calculating the discriminant:
\[ (-8)^2 = 64 \]
\[ 4 \cdot 3 \cdot (-16) = -192 \quad \Rightarrow \quad 64 + 192 = 256 \]
Now substituting back:
\[ x = \frac{8 \pm \sqrt{256}}{6} \]
\[ \sqrt{256} = 16 \]
Thus:
\[ x = \frac{8 \pm 16}{6} \]
Calculating the two possible values:
1. \( x = \frac{24}{6} = 4 \)
2. \( x = \frac{-8}{6} = -\frac{4}{3} \) (not valid since side lengths cannot be negative)

So, we have:
\[ x = 4 \text{ cm} \]

### Step 7: Calculate the perimeter of the smaller square
The perimeter \( P \) of a square is given by:
\[ P = 4 \times \text{side} \]
Thus, for the smaller square:
\[ P = 4 \times x = 4 \times 4 = 16 \text{ cm} \]

### Final Answer
The perimeter of the smaller square is \( 16 \) cm.

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