For the given fixed perimeter of 50 cm, the total number of rectangles which must have its sides in integers (cm) is:

Options

A. 50
B. 25
C. 12
D. infinite

Answer

To find the total number of rectangles with integer side lengths that can be formed with a fixed perimeter of 50 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the perimeter formula for a rectangle**:
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
where \( L \) is the length and \( B \) is the breadth.

2. **Set up the equation with the fixed perimeter**:
Given that the perimeter is 50 cm, we can set up the equation:
\[ 2(L + B) = 50 \]
Dividing both sides by 2 gives:
\[ L + B = 25 \]

3. **Express breadth in terms of length**:
From the equation \( L + B = 25 \), we can express \( B \) as:
\[ B = 25 - L \]

4. **Determine the integer constraints**:
Since both \( L \) and \( B \) must be positive integers, we need:
\[ L > 0 \quad \text{and} \quad B > 0 \]
This implies:
\[ 25 - L > 0 \quad \Rightarrow \quad L < 25 \]
Therefore, \( L \) can take any integer value from 1 to 24.

5. **Count the possible integer values for \( L \)**:
The possible integer values for \( L \) are:
\[ L = 1, 2, 3, \ldots, 24 \]
This gives us a total of 24 possible values for \( L \).

6. **Account for unique rectangles**:
However, since the rectangle with sides \( L \) and \( B \) is the same as the rectangle with sides \( B \) and \( L \), we need to consider only unique pairs. Thus, we only count pairs where \( L \leq B \).

7. **Find the unique pairs**:
The unique pairs can be found by considering:
\[ L + B = 25 \quad \text{and} \quad L \leq B \]
This means we only need to consider values of \( L \) up to 12 (since \( 12 + 13 = 25 \) is the last unique pair). The pairs are:
- \( (1, 24) \)
- \( (2, 23) \)
- \( (3, 22) \)
- \( (4, 21) \)
- \( (5, 20) \)
- \( (6, 19) \)
- \( (7, 18) \)
- \( (8, 17) \)
- \( (9, 16) \)
- \( (10, 15) \)
- \( (11, 14) \)
- \( (12, 13) \)

This gives us a total of 12 unique rectangles.

### Final Answer:
The total number of rectangles with integer sides for a fixed perimeter of 50 cm is **12**.

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