If in a parallelogram, the diagonals are equal and perpendicular, then it is a:-

Question

Accepted Answer

The correct option is 3:

Let's understand the properties of different quadrilaterals, starting with a parallelogram. A parallelogram is a quadrilateral with two pairs of parallel sides. Key properties of a parallelogram include:  * Opposite sides are equal.
  * Opposite angles are equal.
  * Diagonals bisect each other.Now, let's consider the additional conditions given in the problem: the diagonals are equal and perpendicular.1.  **Diagonals are equal:**
    If the diagonals of a parallelogram are equal, then the parallelogram is a rectangle. A rectangle is a parallelogram where all four angles are right angles ($90^\circ$).

2.  **Diagonals are perpendicular:**
    If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. A rhombus is a parallelogram where all four sides are equal in length.When both conditions are met, i.e., the diagonals are equal AND perpendicular, the figure must possess the properties of both a rectangle and a rhombus. A quadrilateral that is both a rectangle and a rhombus is a **square**.

A square has the following properties:  * All four sides are equal (property of a rhombus).
  * All four angles are right angles ($90^\circ$) (property of a rectangle).
  * Diagonals are equal (property of a rectangle).
  * Diagonals bisect each other at right angles (property of a rhombus).Since the problem states that in a parallelogram, the diagonals are equal and perpendicular, it perfectly matches the definition and properties of a square.

The incorrect option is 1: Rhombus
A rhombus is a parallelogram whose diagonals are perpendicular. However, the diagonals of a rhombus are generally not equal unless it is also a square. Since the problem specifies that the diagonals are *equal* and perpendicular, a rhombus alone is not sufficient.

The incorrect option is 2: Rectangle
A rectangle is a parallelogram whose diagonals are equal. However, the diagonals of a rectangle are generally not perpendicular unless it is also a square. Since the problem specifies that the diagonals are equal and *perpendicular*, a rectangle alone is not sufficient.

The incorrect option is 4: Trapezium
A trapezium (or trapezoid) is a quadrilateral with at least one pair of parallel sides. It is a more general category and does not necessarily have equal or perpendicular diagonals, nor does it have the specific properties of a parallelogram that would lead to a square under the given conditions. A trapezium is not necessarily a parallelogram.