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 sides PQ, QR, RS, and SP be A, B, C, and D respectively.

2. **Apply the Midpoint Theorem**:
According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. Therefore:
- \( AB \) is parallel to \( QS \) and \( AB = \frac{1}{2} QS \)
- \( CD \) is parallel to \( PQ \) and \( CD = \frac{1}{2} PQ \)

3. **Consider Triangles**:
We can analyze triangles formed by PQS and PRS:
- In triangle PQS, since A and B are midpoints, we have:
\[ AB = \frac{1}{2} QS \]
- In triangle PRS, since C and D are midpoints, we have:
\[ CD = \frac{1}{2} PR \]

4. **Establish Similarity**:
Since A and B are midpoints, triangle PAB is similar to triangle PQS. This means:
\[ \frac{AB}{QS} = \frac{1}{2} \]
and similarly for triangle PRS:
\[ \frac{CD}{PR} = \frac{1}{2} \]

5. **Set Equal Lengths**:
For the quadrilateral ABCD to be a rhombus, the lengths of opposite sides must be equal. Therefore, we need:
\[ AB = CD \]
From the previous steps, we have:
\[ \frac{1}{2} QS = \frac{1}{2} PR \]
This simplifies to:
\[ QS = PR \]

6. **Conclusion**:
Therefore, the quadrilateral formed by joining the midpoints of the sides of quadrilateral PQRS is a rhombus if and only if the diagonals PR and QS are equal in length.