The lengths of one side and a diagonal of a rectangle are 63 cm and 65 cm, respectively. What is the perimeter (in cm) of a square whose area is one-seventh of that of the rectangle?

Options

A. 36
B. 64
C. 24
D. 48

Answer

To solve the problem, we need to find the perimeter of a square whose area is one-seventh of the area of a rectangle. We are given the lengths of one side and a diagonal of the rectangle.

**Step 1: Identify the dimensions of the rectangle.**
- We know one side of the rectangle (let's call it \( a \)) is 63 cm.
- The diagonal (let's call it \( d \)) is 65 cm.

**Step 2: Use the Pythagorean theorem to find the other side of the rectangle.**
- According to the Pythagorean theorem, for a rectangle, the relationship between the sides and the diagonal is given by:
\[ d^2 = a^2 + b^2 \]
where \( b \) is the other side of the rectangle. Plugging in the values we have:
\[ 65^2 = 63^2 + b^2 \]
\[ 4225 = 3969 + b^2 \]
\[ b^2 = 4225 - 3969 = 256 \]
\[ b = \sqrt{256} = 16 \text{ cm} \]

**Step 3: Calculate the area of the rectangle.**
- The area \( A \) of the rectangle is given by:
\[ A = a \times b = 63 \times 16 \]
\[ A = 1008 \text{ cm}^2 \]

**Step 4: Find one-seventh of the area of the rectangle.**
- Now, we need to find one-seventh of the area:
\[ \text{Area of square} = \frac{1}{7} \times 1008 = 144 \text{ cm}^2 \]

**Step 5: Determine the side length of the square.**
- Let \( s \) be the side length of the square. The area of the square is given by:
\[ s^2 = 144 \]
\[ s = \sqrt{144} = 12 \text{ cm} \]

**Step 6: Calculate the perimeter of the square.**
- The perimeter \( P \) of the square is given by:
\[ P = 4 \times s = 4 \times 12 = 48 \text{ cm} \]

Thus, the perimeter of the square is **48 cm**.

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