P,Q,R and S are the mid points of sides AB,BC , CD and DA respectively of rhombus ABCD. Show that PQRS is a rectangle.
Under what conditions will PQRS be a square ?

Answer

To show that PQRS is a rectangle, we will follow these steps:

### Step 1: Identify the midpoints
Let P, Q, R, and S be the midpoints of sides AB, BC, CD, and DA of rhombus ABCD, respectively.

### Step 2: Use the Midpoint Theorem
According to the Midpoint Theorem, if we take the triangle ACD, the line segment SR (joining midpoints S and R) is parallel to the diagonal AC and half its length. Thus, we can write:
\[ SR \parallel AC \quad \text{and} \quad SR = \frac{1}{2} AC \]

### Step 3: Apply the Midpoint Theorem to triangle ABC
Similarly, in triangle ABC, the line segment PQ (joining midpoints P and Q) is parallel to the diagonal AC and half its length. Thus, we can write:
\[ PQ \parallel AC \quad \text{and} \quad PQ = \frac{1}{2} AC \]

### Step 4: Establish that PQ and SR are equal and parallel
From Steps 2 and 3, we have:
- \( PQ \parallel SR \)
- \( PQ = SR \)

### Step 5: Show that opposite sides are equal and parallel
Since PQ and SR are equal and parallel, we can conclude that:
- \( PQ \parallel SR \)
- \( PS \parallel QR \) (by similar reasoning using triangles ABD and BCD)

### Step 6: Show that PQRS is a parallelogram
Since both pairs of opposite sides (PQ and SR, PS and QR) are equal and parallel, we can conclude that PQRS is a parallelogram.

### Step 7: Show that PQRS is a rectangle
In a rhombus, the diagonals bisect each other at right angles. Therefore, the angles formed by the diagonals (like angle O in the midpoint quadrilateral) are 90 degrees. Since one angle of parallelogram PQRS is 90 degrees, it follows that:
- PQRS is a rectangle.

### Conclusion for Rectangle
Thus, we have shown that PQRS is a rectangle.

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### Conditions for PQRS to be a Square
To determine under what conditions PQRS becomes a square, we need to analyze the triangles formed by the diagonals.

### Step 8: Analyze triangles POS and ROS
Join points P and R, and points O and S. In triangles POS and ROS, we have:
- OS is common to both triangles.
- O is the center of the rhombus, so OP = OR.

### Step 9: Use the criteria for congruence
For triangles POS and ROS to be congruent, the angles POS and ROS must be equal. Since these angles form a linear pair, they are equal if and only if they are both 90 degrees.

### Step 10: Condition for angles
Thus, for PQRS to be a square, we need:
- Angle POS = Angle ROS = 90 degrees.

### Conclusion for Square
This condition is satisfied if and only if rhombus ABCD is a square. Therefore, PQRS is a square if and only if ABCD is a square.

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