In each of the following questions, a is followed by information given in three statements. You have to decide the information given in which of the statements is necessary and sufficient to answer the question.
What is the measurement of perimeter of a semicircle in cm ?
(1) Area of the semicircle is equal to the area of the parallelogram.
(2) Length of the parallelogram is 1.5 times of the radius of the semicircle.
(3) The difference of the length and breadth of a parallelogram is 8 cm.

Options

A. Only (2) and (3) are sufficient
B. Only (1) and (3) are sufficient
C. Only (1) and (2) are sufficient
D. (1),(2) and (3) together are not sufficient to answer the question.

Answer

To determine the measurement of the perimeter of a semicircle, we need to analyze the provided statements and see if they give us the necessary information to find the radius of the semicircle, which is essential for calculating the perimeter.

### Step-by-Step Solution:

1. **Understanding the Perimeter of a Semicircle**:
The perimeter \( P \) of a semicircle can be calculated using the formula:
\[ P = \pi r + 2r \]
where \( r \) is the radius of the semicircle. Thus, we need to find the value of \( r \).

2. **Analyzing the Statements**:
- **Statement 1**: The area of the semicircle is equal to the area of the parallelogram.
- The area of a semicircle is given by:
\[ \text{Area}_{\text{semicircle}} = \frac{1}{2} \pi r^2 \]
- The area of a parallelogram is given by:
\[ \text{Area}_{\text{parallelogram}} = \text{length} \times \text{breadth} \]
- This statement alone does not provide enough information about the dimensions of the parallelogram to find \( r \).

- **Statement 2**: The length of the parallelogram is 1.5 times the radius of the semicircle.
- Let the length of the parallelogram be \( l = 1.5r \).
- This gives a relationship between the length and the radius, but we still need the breadth to find the area of the parallelogram.

- **Statement 3**: The difference between the length and breadth of the parallelogram is 8 cm.
- Let the breadth be \( b \). Then we have:
\[ l - b = 8 \quad \Rightarrow \quad 1.5r - b = 8 \quad \Rightarrow \quad b = 1.5r - 8 \]
- Now we have expressions for both the length and breadth in terms of \( r \).

3. **Combining Statements**:
- From Statement 1, we can set up the equation:
\[ \frac{1}{2} \pi r^2 = l \times b = (1.5r) \times (1.5r - 8) \]
- This equation can be solved for \( r \).

4. **Conclusion**:
- All three statements together provide sufficient information to calculate the radius \( r \) of the semicircle.
- Once we have \( r \), we can substitute it back into the perimeter formula to find the perimeter of the semicircle.

### Final Answer:
The necessary and sufficient statements to determine the perimeter of the semicircle are **all three statements together**.