The quadrilateral formed by joining the mid-points of the side for quadilateral PQRS, taken in order, is a rhombus, if

Options

A. PQRS is a rhombus
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 rhombus, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Midpoints**:
Let the midpoints of the sides of quadrilateral PQRS be A, B, C, and D, where:
- A is the midpoint of PQ
- B is the midpoint of QR
- C is the midpoint of RS
- D is the midpoint of SP

2. **Use the Midpoint Theorem**:
According to the midpoint theorem, if we take any triangle, the line segment joining the midpoints of any two sides is parallel to the third side and half as long.
- In triangle PSQ, since A and D are midpoints, we have:
\[ AD = \frac{1}{2} SQ \]
- In triangle PQR, since A and B are midpoints, we have:
\[ AB = \frac{1}{2} PR \]

3. **Establish Relationships**:
Since we want quadrilateral ABCD to be a rhombus, all sides must be equal:
\[ AB = BC = CD = DA \]

4. **Equate the Lengths**:
From the relationships established using the midpoint theorem:
- From triangle PSQ:
\[ AD = \frac{1}{2} SQ \quad \text{(1)} \]
- From triangle PQR:
\[ AB = \frac{1}{2} PR \quad \text{(2)} \]
- Setting these equal gives:
\[ \frac{1}{2} SQ = \frac{1}{2} PR \]
- Simplifying this leads to:
\[ SQ = PR \]

5. **Conclusion**:
The diagonals of quadrilateral PQRS must be equal for quadrilateral ABCD (formed by joining the midpoints) to be a rhombus. Thus, the correct condition is:
- **The diagonals of PQRS are equal**.

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