The following questions are accompanied by three statements A, B and C. You have to determine which statement(s) is/are necessary/sufficient to answer the question.
Find area of a rectangle.
A. Length and breadth of the rectangle are in the ratio of 4:3.
B. Sum of the lengths of diagonals of the rectangle is 50 m.
C. Area of a square is `1225 m^(2),` whose perimeter is twice the perimeter of the rectangle.

Options

A. Only A and B together
B. Only A and C together
C. All the three together
D. Any two of the three together

Answer

To determine the area of a rectangle, we need to find the product of its length and breadth. Let's analyze the given statements one by one to see if they provide sufficient information to calculate the area.

### Step 1: Analyze Statement A
**Statement A:** The length and breadth of the rectangle are in the ratio of 4:3.

- This statement gives us a ratio, but we do not have the actual lengths of the length and breadth.
- We can express the length as \(4x\) and the breadth as \(3x\) for some value of \(x\).
- However, we cannot calculate the area without knowing the actual value of \(x\).

**Conclusion for A:** Insufficient alone.

### Step 2: Analyze Statement B
**Statement B:** The sum of the lengths of the diagonals of the rectangle is 50 m.

- The diagonals of a rectangle are equal. Therefore, if the sum of the lengths of the diagonals is 50 m, each diagonal is \(25\) m.
- The length of the diagonal \(d\) can be expressed using the Pythagorean theorem:
\[ d = \sqrt{l^2 + b^2} \]
where \(l\) is the length and \(b\) is the breadth.
- Knowing \(d = 25\), we have:
\[ \sqrt{l^2 + b^2} = 25 \implies l^2 + b^2 = 625 \]
- However, without knowing either \(l\) or \(b\) specifically, we cannot determine the area.

**Conclusion for B:** Insufficient alone.

### Step 3: Analyze Statement C
**Statement C:** The area of a square is \(1225 \, m^2\), whose perimeter is twice the perimeter of the rectangle.

- The area of the square gives us the side length:
\[ \text{Side} = \sqrt{1225} = 35 \, m \]
- The perimeter of the square is:
\[ \text{Perimeter} = 4 \times 35 = 140 \, m \]
- Since the perimeter of the rectangle is half of this, we have:
\[ \text{Perimeter of rectangle} = \frac{140}{2} = 70 \, m \]
- The perimeter of a rectangle is given by \(2(l + b) = 70\), so:
\[ l + b = 35 \]

### Step 4: Combine Statements A and C
From Statement A, we have the ratio \(l:b = 4:3\). Let:
- \(l = 4x\)
- \(b = 3x\)

Using the equation from Statement C:
\[ 4x + 3x = 35 \implies 7x = 35 \implies x = 5 \]

Now substituting \(x\) back:
- \(l = 4x = 20 \, m\)
- \(b = 3x = 15 \, m\)

### Step 5: Calculate the Area
Now we can calculate the area of the rectangle:
\[ \text{Area} = l \times b = 20 \times 15 = 300 \, m^2 \]

### Conclusion
To find the area of the rectangle, we used:
- Statement A (for the ratio of length and breadth)
- Statement C (for the perimeter)

Thus, the area of the rectangle is \(300 \, m^2\), and we conclude that **Statements A and C together are sufficient** to answer the question.