Is quadrilateral ABCD a rhombus?

(1) Line segments AC and BD are perpendicular bisectors of each other.

(2) AB = BC = CD = AD


I started with option(2) and AB=BC=CD=AD holds true for either Square or Rhombus..So, insufficient

From option(1), again the condition holds true for Square as well as for Rhombus [ Diagonals are perpendicular bisector ]

Combining both the statement, again left with Square or Rhombus. So, my answer was (E).

But the OA says (D) ..Am I missing something, please help !!

FACT - Every square is a rhombus. So any quadrilateral satisfying (1) or (2) will at least be a rhombus. Even if it is a square, it is a rhombus too.

Hope it helps

GMAT 2007

I made the same mistake and thought answer is (E). But after reviewing definitions of Rhombus and square in some geometry books, this is what I inferred from that.

1) Square is a special type of Rhombus OR The set of squares is a subset of the set of rhombuses.
2) Rhombus has all the 4 sides equal.
3) The diagonals of rhombus are perpendicular bisectors of each other. This condition is not true for a parallelogram or any other quadrilateral other than square. For a rectangle, the diagonals bisect, but they are not perpendicular.
4) A Square is a special type of rhombus in which the angles are 90 degrees and they have the same property of Rhombus - that is its diagonals bisect each other and they are also perpendicular.

So if you read Statement (1), it says the diagonals are perpendicular bisectors to each other. Based on the above definition, we can certainly say that this is a Rhombus. But we cannot say it is a Square, because we don't know whether the angles of the quadrilateral are 90 degrees each. So to answer whether it is a Rhombus, this statement is sufficient.

Statement (2) says AB = BC = CD = DA, which means all sides are equal. Again we can certainly say that the quadrilateral is a Rhombus, but we cannot say that it is a Square.

So the answer is (D).

Hope this helps.

Another great explanation, Harish Dorai.

Thanks,

Jad