please verify :- squares and rhombus

[divyagokulnath]

In this Data Suff question, which is there in the geometry question bank

Is quadrilateral ABCD a rhombus?

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

(2) AB = BC = CD = AD

The answer provided in option (d), which means that either statement alone suffices.
However, even in a square, aren't the diagonals perpendicular to each other?
how can u determine whether the question is about a rhombus or a square?

similarly, from statement 2, AB=BC=CD=AD can happen either in a square or a rhombus

The statement needed is whether the angles are 90 degrees or not...

please verify this question
regards
divya

[divyagokulnath]

ok i viewed the reply in one of the other posts :) so every square is a rhombus..

[Ben Ku]

Yea, students get rhombuses (actually, rhombi) and squares confused all the time, because rhombi are not mentioned much in school.

A rhombus is any quadrailateral with four congruent sides. By that definition, a square is a special type of rhombus (with four congruent angles).

Let me know if there are additional questions.

[Ben Ku]

In both the kite and rhombus, the diagonals are perpendicular. However, because the rhombus is a parallelogram, the diagonals also must bisect each other. This is not true for the kite because the kite is not a parallelogram.

(1) is fine because it says the diagonals are "perpendicular bisectors."

[chitrangada.maitra]

So, its safe to say that the following properties are true:

1. If a quadrilateral has perpendicular bisectors, their opposite angles are equal

2. If the four sides of any quadrilateral are equal, the opposite angles of the quadrilateral will be equal as well.

Because, if the above mentioned does not hold true in all cases, then the quadrilateral cannot be a rhombus.

[mschwrtz]

Those two rules are right chitrangada.maitra, but let's make them a bit stronger:

1. The diagonals of a quadrilateral are perpendicular bisectors IF AND ONLY IF its opposite angles are equal.

2. The four sides of any quadrilateral are equal IF AND ONLY IF the opposite angles of the quadrilateral are equal.

By the way, if you can bear with a little math nerdiness, a rhombus is a special kite (i.e., every rhombus is a kite, but not every kite is a rhombus). The diagonals of a rhombus are perpendicular, just like the diagonals of every other kite.

A rhombus is also a special parallelogram. the diagonals of a rhombus bisect one another, just as do the diagonals of every other parallelogram.

If you have a figure that's both a rectangle and a kite, you have a rhombus.