Parallelogram Vs Rhombus Vs Square

[karthikpandian19]

E, F, G, and H are the vertices of a polygon. Is polygon EFGH a square?

(1) EFGH is a parallelogram.

(2) The diagonals of EFGH are perpendicular bisectors of one another.

.....
I got confused with the properties even though i knew what each are individually. The relative properties between the 3 messed me around. Nice question though

[Anurag@Gurome]

E, F, G, and H are the vertices of a polygon. Is polygon EFGH a square?

(1) EFGH is a parallelogram.
(2) The diagonals of EFGH are perpendicular bisectors of one another.

To be a square a polygon must have four equal sides and four equal angles, i.e. four right angles.

Statement 1: EFGH may or may not be a square.

Not sufficient

Statement 2: As the diagonals of EFGH are perpendicular bisector of one another, EFGH must be a rhombus. A rhombus is a polygon with four equal sides and opposite sides are parallel to each other. But EFGH may or may not be a square.

Not sufficient

1& 2 Together: Both statements together do not add any new information as any rhombus is essentially a parallelogram.

Not sufficient

The correct answer is E.

[Anaira Mitch]

To prove that a quadrilateral is a square, you must prove that it is both a rhombus (all sides are equal) and a rectangle (all angles are equal).

(1) INSUFFICIENT: Not all parallelograms are squares (however all squares are parallelograms).

(2) INSUFFICIENT: If a quadrilateral has diagonals that are perpendicular bisectors of one another, that quadrilateral is a rhombus. Not all rhombuses are squares (however all squares are rhombuses).

If we look at the two statements together, they are still insufficient. Statement (2) tells us that ABCD is a rhombus, so statement one adds no more information (all rhombuses are parallelograms). To prove that a rhombus is a square, you need to know that one of its angles is a right angle or that its diagonals are equal (i.e. that it is also a rectangle).

The correct answer is E