The quadrilateral formed by joining the mid-points of the sides of a quadrilateral PQRS, taken in order, is a rectangle, if

Options

A. PQRS is a rectangle
B. PQRS is parallelogram
C. diagonals of PQRS are perpendicular
D. diagonals of PQRS are equal

Explanation

To determine the condition under which the quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS is a rectangle, we can analyze the properties of the quadrilateral and its midpoints step by step.

### Step-by-Step Solution:

1. **Identify the Midpoints**:
- Let A, B, C, and D be the midpoints of sides PQ, QR, RS, and SP respectively.
- Therefore, A = midpoint of PQ, B = midpoint of QR, C = midpoint of RS, D = midpoint of SP.

2. **Form the Quadrilateral**:
- Join the points A, B, C, and D to form the quadrilateral ABCD.

3. **Properties of a Rectangle**:
- For ABCD to be a rectangle, we need to show that the diagonals AC and BD are equal and that they bisect each other at right angles.

4. **Analyze the Diagonals**:
- In a rectangle, the diagonals are equal. Therefore, we need to check the lengths of diagonals AC and BD.
- From the properties of midpoints, we know that the length of diagonal AC is equal to half the length of diagonal PR of quadrilateral PQRS, and the length of diagonal BD is equal to half the length of diagonal QS of quadrilateral PQRS.

5. **Condition for Equality of Diagonals**:
- For ABCD to be a rectangle, we need the diagonals of quadrilateral PQRS (PR and QS) to be equal. This is because if PR = QS, then AC = BD, satisfying the condition for ABCD to be a rectangle.

6. **Check for Perpendicularity**:
- Additionally, for ABCD to be a rectangle, the diagonals must also be perpendicular. This can happen if the diagonals of the original quadrilateral PQRS are perpendicular to each other.

7. **Conclusion**:
- Therefore, the quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS is a rectangle if the diagonals of PQRS are equal and/or perpendicular.

### Final Answer:
The quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS is a rectangle if the diagonals of PQRS are perpendicular.