Why is this statement NOT always true?:
The longer diagonal of the rhombus is perpendicular to two sides of the rhombus.
I'd say it's never true.
The diagonals of a rhombus are perpendicular bisectors of each other. So, unless the shorter diagonal is parallel to a side, the longer cannot be perpendicular. Since the diagonals are transversals between parallel sides, they cannot pe parallel to them.
To understand why the statement "The longer diagonal of the rhombus is perpendicular to two sides of the rhombus" is not always true, we first need to define what a rhombus is and understand its properties.
A rhombus is a quadrilateral with all sides of equal length. In a rhombus, opposite angles are equal, but opposite sides are not necessarily parallel. The diagonals of a rhombus bisect each other at right angles, meaning they intersect at a 90-degree angle.
Now, let's consider a specific case where the longer diagonal is not perpendicular to two sides of a rhombus. Imagine a rhombus with one acute angle. One of the diagonals of this rhombus will be longer, while the other will be shorter.
In this case, the longer diagonal will not be perpendicular to two sides of the rhombus. Instead, it will be perpendicular to only one side of the rhombus. The shorter diagonal, on the other hand, will be perpendicular to the remaining side. This scenario breaks the statement since it is not always true that the longer diagonal is perpendicular to two sides of a rhombus.
Therefore, we can conclude that the statement is not always true because the longer diagonal of a rhombus is not necessarily perpendicular to two sides.