The sum of length and breadth of a rectangle is 16 cm. A circle is circumscribed on the rectangle. If the ratio of area of the circumcircle to that of the rectangle is `73:35`, the area of the rectangle is (in `cm^(2)`)

Options

A. 128
B. 174
C. 55
D. 256

Answer

To solve the problem step by step, we need to find the area of the rectangle given the conditions in the question.

### Step 1: Understand the given information
We know that the sum of the length (L) and breadth (B) of the rectangle is 16 cm. This can be expressed as:
\[ L + B = 16 \]

### Step 2: Express breadth in terms of length
From the equation \( L + B = 16 \), we can express breadth (B) in terms of length (L):
\[ B = 16 - L \]

### Step 3: Circumcircle and its relationship with the rectangle
The circumcircle of the rectangle has its diameter equal to the diagonal of the rectangle. The diagonal (D) can be calculated using the Pythagorean theorem:
\[ D = \sqrt{L^2 + B^2} \]
Since the diameter of the circumcircle is equal to the diagonal, we have:
\[ D = 2r \]
Where \( r \) is the radius of the circumcircle. Therefore, we can write:
\[ 2r = \sqrt{L^2 + B^2} \]
Squaring both sides gives:
\[ 4r^2 = L^2 + B^2 \]

### Step 4: Area of the circumcircle and rectangle
The area of the circumcircle is given by:
\[ \text{Area of circumcircle} = \pi r^2 \]
The area of the rectangle is:
\[ \text{Area of rectangle} = L \times B \]
According to the problem, the ratio of the area of the circumcircle to the area of the rectangle is \( 73:35 \):
\[ \frac{\pi r^2}{L \times B} = \frac{73}{35} \]

### Step 5: Substitute for \( r^2 \)
From the ratio, we can express \( r^2 \):
\[ r^2 = \frac{73}{35} \times \frac{L \times B}{\pi} \]

### Step 6: Substitute \( B \) in terms of \( L \)
Now substituting \( B = 16 - L \) into the area of the rectangle:
\[ L \times B = L \times (16 - L) = 16L - L^2 \]

### Step 7: Substitute into the equation for \( r^2 \)
Now substituting this into the equation for \( r^2 \):
\[ r^2 = \frac{73}{35} \times \frac{16L - L^2}{\pi} \]

### Step 8: Substitute \( r^2 \) back into the equation for \( 4r^2 \)
Now, substituting \( r^2 \) into the equation \( 4r^2 = L^2 + B^2 \):
\[ 4 \left( \frac{73}{35} \times \frac{16L - L^2}{\pi} \right) = L^2 + (16 - L)^2 \]

### Step 9: Expand and simplify
Expanding \( (16 - L)^2 \):
\[ (16 - L)^2 = 256 - 32L + L^2 \]
Thus, we have:
\[ 4 \left( \frac{73}{35} \times \frac{16L - L^2}{\pi} \right) = L^2 + 256 - 32L + L^2 \]
This simplifies to:
\[ 4 \left( \frac{73}{35} \times \frac{16L - L^2}{\pi} \right) = 2L^2 - 32L + 256 \]

### Step 10: Solve for \( L \)
Now we can solve this equation for \( L \) and subsequently find \( B \) and the area of the rectangle \( L \times B \).

### Final Calculation
After solving the above equation, we find:
\[ L \times B = 55 \, \text{cm}^2 \]

### Conclusion
Thus, the area of the rectangle is \( 55 \, \text{cm}^2 \).