Is it correct to say that a parallelogram having perpendicular distance between the opposite sides equal, is a rhombus ? If yes, then how ?
Yes, it is correct to say that a parallelogram with perpendicular distance between the opposite sides equal is a rhombus. To understand why, let's go through the properties of a rhombus and how they relate to the given condition.
1. Opposite sides are parallel: This is a property of all parallelograms, including a rhombus. In a parallelogram, the opposite sides are parallel, meaning they never intersect.
2. Opposite sides are equal in length: This is also a property of all parallelograms, including a rhombus. In a parallelogram, the opposite sides are congruent (have the same length).
3. Perpendicular diagonals: A rhombus has diagonals that intersect at right angles. Since a diagonal of a parallelogram divides it into two congruent triangles, the fact that the diagonals of a rhombus are perpendicular means that the triangles formed are right triangles. In other words, the diagonals are the altitudes of these triangles.
Now, let's consider the condition given in the question: a parallelogram with perpendicular distance between the opposite sides equal.
In a rhombus, the diagonals are perpendicular, which means they intersect at a right angle. Since the diagonals bisect each other (they divide each other into two equal parts), the intersection point of the diagonals is equidistant from all four sides of the rhombus.
Conversely, if we have a parallelogram where the perpendicular distance between the opposite sides is equal, it means that the diagonals must be perpendicular and bisect each other. Therefore, this parallelogram has all the properties of a rhombus, making it a rhombus.
In summary, a parallelogram with perpendicular distance between the opposite sides equal is a rhombus because it satisfies all the necessary conditions for a rhombus: opposite sides parallel, opposite sides equal in length, and perpendicular diagonals.