A circular wire of diameter 112 cm is cut and bent in the form of a rectangle whose sides are in the ratio of 9 : 7 The smaller side of the rectangle is.
112 cm cm ब्याज का एक वृत्ताकार तार आयत के आकार में काटा जाता है जिसकी भुजाएं 9 : 7 के अनुपात में है। आयत की छोटी भुजा कितनी होगी?

Options

A. 87 cm
B. 77 cm
C. 97 cm
D. 67 cm

Explanation

To solve the problem, we need to find the smaller side of a rectangle formed by bending a circular wire of diameter 112 cm. The sides of the rectangle are in the ratio of 9:7.

### Step-by-Step Solution:

1. **Calculate the Circumference of the Circular Wire:**
The circumference \( C \) of a circle is given by the formula:
\[ C = \pi \times D \]
where \( D \) is the diameter of the circle. Given \( D = 112 \) cm, we can use \( \pi \approx \frac{22}{7} \) for our calculations.
\[ C = \frac{22}{7} \times 112 \]

2. **Perform the Multiplication:**
First, simplify the multiplication:
\[ C = \frac{22 \times 112}{7} \]
To calculate \( 22 \times 112 \):
\[ 22 \times 112 = 2464 \]
Now, divide by 7:
\[ C = \frac{2464}{7} = 352 \text{ cm} \]

3. **Set Up the Perimeter of the Rectangle:**
The perimeter \( P \) of the rectangle is equal to the circumference of the circle:
\[ P = 2(L + B) \]
where \( L \) and \( B \) are the length and breadth of the rectangle. Given the ratio of the sides is 9:7, we can express:
\[ L = 9x \quad \text{and} \quad B = 7x \]

4. **Substitute into the Perimeter Formula:**
Substitute \( L \) and \( B \) into the perimeter formula:
\[ 2(9x + 7x) = 352 \]
Simplifying gives:
\[ 2(16x) = 352 \]
\[ 32x = 352 \]

5. **Solve for \( x \):**
Divide both sides by 32:
\[ x = \frac{352}{32} = 11 \]

6. **Calculate the Smaller Side:**
The smaller side \( B \) is given by:
\[ B = 7x = 7 \times 11 = 77 \text{ cm} \]

### Conclusion:
The smaller side of the rectangle is **77 cm**.