In a quadrilateral PQRS, diagonals PR and QS bisect each other at M. If `angle PMS =90^(@)`, then quadrilateral PQRS is a_____.

Options

A. Trapezium
B. Kite
C. Rhombus
D. Parallelogram

Answer

To solve the problem, we need to analyze the properties of the quadrilateral PQRS based on the information given.

### Step-by-Step Solution:

1. **Understanding the Quadrilateral**:
We have a quadrilateral PQRS with diagonals PR and QS that bisect each other at point M. This means that point M is the midpoint of both diagonals.

2. **Given Information**:
We know that the angle PMS is 90 degrees (∠PMS = 90°). This indicates that the diagonals intersect at a right angle.

3. **Properties of Quadrilaterals**:
We need to recall the properties of different types of quadrilaterals:
- **Trapezium**: Only one pair of opposite sides is parallel; diagonals do not necessarily bisect each other.
- **Kite**: Two pairs of adjacent sides are equal; diagonals are perpendicular but do not necessarily bisect each other.
- **Rhombus**: All sides are equal; diagonals bisect each other at right angles.
- **Parallelogram**: Opposite sides are equal and parallel; diagonals bisect each other but not necessarily at right angles.

4. **Analyzing the Given Conditions**:
Since the diagonals bisect each other at M and ∠PMS = 90°, this matches the properties of a rhombus, where the diagonals bisect each other at right angles.

5. **Conclusion**:
Therefore, based on the properties of quadrilaterals and the given conditions, we can conclude that quadrilateral PQRS is a **Rhombus**.

### Final Answer:
Quadrilateral PQRS is a **Rhombus**.