The area of a square of one side 8 cm is equal to the area of a rectangle. Which of the following statements about the rectangle is/are correct?
The length of the rectangle is 16 times of the breadth
The length of the rectangle is 32 times of the breadth
The breadth of the rectangle is `1/6` of the length.

Options

A. Only (1)
B. Only (3) and (4)
C. Either (1) or (2)
D. Either (3) or (4)

Answer

To solve the problem, we need to find out which of the statements regarding the rectangle is correct based on the area of a square.

### Step 1: Calculate the area of the square
The area of a square is given by the formula:
\[ \text{Area} = \text{side}^2 \]
Given that the side of the square is 8 cm, we calculate:
\[ \text{Area} = 8^2 = 64 \text{ cm}^2 \]

### Step 2: Set the area of the rectangle equal to the area of the square
Since the area of the rectangle is equal to the area of the square, we have:
\[ \text{Area of rectangle} = \text{Length} \times \text{Breadth} = 64 \text{ cm}^2 \]

### Step 3: Analyze the statements
We will analyze each statement one by one.

#### Statement 1: The length of the rectangle is 16 times the breadth.
Let the breadth be \( b \). Then the length \( l \) can be expressed as:
\[ l = 16b \]
Substituting into the area equation:
\[ l \times b = 64 \implies 16b \times b = 64 \implies 16b^2 = 64 \implies b^2 = 4 \implies b = 2 \text{ cm} \]
Now, substituting back to find \( l \):
\[ l = 16 \times 2 = 32 \text{ cm} \]
This statement is correct.

#### Statement 2: The length of the rectangle is 32 times the breadth.
Using the same notation, if \( l = 32b \):
\[ l \times b = 64 \implies 32b \times b = 64 \implies 32b^2 = 64 \implies b^2 = 2 \implies b = \sqrt{2} \text{ cm} \]
Now, substituting back to find \( l \):
\[ l = 32 \times \sqrt{2} \text{ cm} \]
This does not yield a whole number for the breadth, and thus this statement is not correct.

#### Statement 3: The breadth of the rectangle is \( \frac{1}{6} \) of the length.
Let the length be \( l \) and the breadth be \( b \):
\[ b = \frac{1}{6}l \]
Substituting into the area equation:
\[ l \times b = 64 \implies l \times \frac{1}{6}l = 64 \implies \frac{1}{6}l^2 = 64 \implies l^2 = 384 \implies l = \sqrt{384} \text{ cm} \]
Calculating \( b \):
\[ b = \frac{1}{6} \sqrt{384} \text{ cm} \]
Again, this does not yield a simple relationship that confirms the area condition, hence this statement is not correct.

### Conclusion
Only the first statement is correct: **The length of the rectangle is 16 times the breadth.**

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