Question

The diagonals of the quadrilateral are perpendicular to each other. Is such a quadrilateral always a rhombus? Draw a rough figure to justify your answer

Answer 1

No, if the diagonals of the quadirateral are perpendicular to each other then such quadirateral is not always a rhombus.
it may be , $\bf{square}$, $\bf{rectangle}$

as you know conditions of rhombus
1. it should be parallelogram.
2. all sides should be equal.
3. diagonals are perpendicular to each other.

but here given only " diagonals of the quadirateral are perpendicular to each other."
hence, we can't say given quadirateral is always rhombus.

Answer 2

In order to solve this question we must know the properties of a rhombus.

A rhombus has 3 main properties:

1. All its sides will be congruent.

2. The diagonals bisect the angles.

3. Diagonals are perpendicular bicestor of each other.

Now coming to the question it is asked if the diagonals of the quadrilateral are perpendicular then will it always be a rhombus then the answer is definitely NO.

The reason is very simple, in case of a square or rectangle the diagonals are perpendicular. Therefore it is not only rhombus having perpendicular diagonals.