Consider the following statements
1. The bisectors of all the four angles of a parallelogram enclose a rectangle.
2. The figure formed by joining the midpoints of the adjacent sides of a rectangle is a rhombus.
3. The figure formed by joining the midpoints of the adjacent sides of a rhombus is a square.
Which of these statements are correct ?

Options

A. 1 and 2
B. 2 and 3
C. 3 and 1
D. 1, 2 and 3

Answer

To solve the problem, we need to analyze each of the three statements provided regarding geometric shapes. Let's break it down step by step.

### Step 1: Analyze Statement 1
**Statement 1:** The bisectors of all the four angles of a parallelogram enclose a rectangle.

- **Explanation:** In a parallelogram, when you bisect each of the four angles, the intersection points of these angle bisectors form a rectangle. This is a known property of parallelograms.

**Conclusion:** Statement 1 is **True**.

### Step 2: Analyze Statement 2
**Statement 2:** The figure formed by joining the midpoints of the adjacent sides of a rectangle is a rhombus.

- **Explanation:** If you take a rectangle and find the midpoints of each side, then connect these midpoints, you will form a rhombus. This is because all sides of the rhombus formed will be equal, as the diagonals of the rectangle are equal and bisect each other.

**Conclusion:** Statement 2 is **True**.

### Step 3: Analyze Statement 3
**Statement 3:** The figure formed by joining the midpoints of the adjacent sides of a rhombus is a square.

- **Explanation:** When you take a rhombus and join the midpoints of its adjacent sides, the figure formed is not a square; it is actually a rectangle. This is because the opposite sides of the rhombus are equal, but the angles are not necessarily 90 degrees.

**Conclusion:** Statement 3 is **False**.

### Final Conclusion
After analyzing all three statements:
- Statement 1 is True.
- Statement 2 is True.
- Statement 3 is False.

Therefore, the correct statements are **1 and 2**.

### Answer:
The correct statements are **1 and 2**.

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