If the diagonals of a quadrilateral bisect one another at right angles, then the quadrilateral is :

Question

Accepted Answer

The correct answer is Option 2: Rhombus.

1. **Explanation for Option 2 (Rhombus):**
   A rhombus is a special type of parallelogram where all sides are of equal length. One of the defining properties of a rhombus is that its diagonals bisect each other at right angles (90 degrees). This means that the diagonals of a rhombus not only cut each other into two equal parts but also intersect at a perpendicular angle. This property is unique to rhombuses among quadrilaterals, which is why Option 2 is the correct answer.

2. **Explanation for why other options are incorrect:**

- **Option 1 (Trapezium):**
     A trapezium (or trapezoid in some regions) is a quadrilateral with at least one pair of parallel sides. The diagonals of a trapezium do not necessarily bisect each other, and they do not intersect at right angles. Therefore, a trapezium does not satisfy the condition given in the question.

- **Option 3 (Parallelogram):**
     While a parallelogram has diagonals that bisect each other, they do not generally intersect at right angles. Only in special cases, such as a rhombus or a rectangle, do the diagonals intersect at right angles. Since the question specifies right angles, a general parallelogram does not meet this criterion.

- **Option 4 (Rectangle):**
     A rectangle is a type of parallelogram where the diagonals bisect each other and are equal in length. However, they do not intersect at right angles. The diagonals of a rectangle intersect at the center, forming two pairs of equal angles, but these angles are not 90 degrees. Therefore, a rectangle does not satisfy the condition of diagonals intersecting at right angles.

In conclusion, the unique property of a rhombus having diagonals that bisect each other at right angles makes Option 2 the correct answer.