Question 1073145: Prove that parallelogram is a rhombus if its diagonals bisect at right angles
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! The diagonals of a parallelogram bisect each other,
splitting the parallelogram into 4 triangles.
If they intersect at right angles,
those 4 triangles are right triangles.
If the length of the diagonals are D and d,
All 4 of those right triangles have legs of length
D/2 and d/2.
With a pair of congruent sides flanking congruent right angles,
they are congruent right triangles,
and their corresponding other sides (hypotenuses)
are also congruent.
So, the 4 sides of the parallelogram (the 4 hypotenuses)
are congruent, and a parallelogram with 4 congruent sides
is called a rhombus.
(It could even be the particular type of rhombus that we call a square).
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