Mgmat quant prob set mistake?????- geometry quad qs

[asth678]

Is quadrilateral ABCD a rhombus?

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

(2) AB = BC = CD = AD

It says the ans is d
but it should be E- cause you can't eliminate that the quad could be a square, you need more info?
Or i m over analyzing it?

[atul.prasad]

I guess the answer should be C..

1) It may not be a rhombus for the following:
http://postimage.org/image/3odchhok/

2) All rhombus are parallelograms. This statement does not confirm that

Together:
http://postimage.org/image/3z116090/

Its given that AB = AD and Angle AOD=AOB is 90 degree

We can prove that all 4 triangles are congruent to each other.

-Opposite angles of a rhombus have equal measure.
-The two diagonals of a rhombus are perpendicular.

From above we can conclude that it is a rhombus

[tim]

Atul, both of your diagrams are incorrect representations of the information given in the statements. The answer of course is D, and you need little more than the definitions of "rhombus" and "perpendicular bisector". 2 is trivially sufficient, because the definition of a rhombus is that all 4 sides are the same length. With statement 1, the diagonals split the figure into four triangles, all of which are right triangles (definition of "perpendicular") and all of which have congruent legs (definition of "bisector"). Thus all four triangles are congruent and so in particular their hypotenuses are all the same, again fulfilling the definition of "rhombus". The issue of whether this is a square is irrelevant, as all squares are automatically rhombuses as well..

[griffin.811]

Regarding Tim's post above, is it necessarily the case that if a quadrilateral has 4 equal sides that it's 4 angles will be congruent to their respective opposing side?

For instance, the geometry strategy guide says the definition of a Rhombus is that all sides are equal, AND, all opposing angles are equal. Stmt (2) however only tells us that all the sides are equal, which leads me to believe that a quadrilateral with 4 equal sides means that all opposing sides have equal angles, and vice versa.

Thanks

[jnelson0612]

Regarding Tim's post above, is it necessarily the case that if a quadrilateral has 4 equal sides that it's 4 angles will be congruent to their respective opposing side?

For instance, the geometry strategy guide says the definition of a Rhombus is that all sides are equal, AND, all opposing angles are equal. Stmt (2) however only tells us that all the sides are equal, which leads me to believe that a quadrilateral with 4 equal sides means that all opposing sides have equal angles, and vice versa.

Thanks

Yes, that is true.

[RonPurewal]

Regarding Tim's post above, is it necessarily the case that if a quadrilateral has 4 equal sides that it's 4 angles will be congruent to their respective opposing side?

For instance, the geometry strategy guide says the definition of a Rhombus is that all sides are equal, AND, all opposing angles are equal. Stmt (2) however only tells us that all the sides are equal, which leads me to believe that a quadrilateral with 4 equal sides means that all opposing sides have equal angles, and vice versa.

Thanks

You're correct, of course (assuming you meant "angle" for the purple word"”"”the current wording is not a thing that actually exists).

You're actually touching on a philosophical issue"”"”namely, the purpose of definitions. That purpose, like the definitions themselves, differs hugely between (a) mathematicians and (b) everyone else.

"- For mathematicians, the goal of a definition is to be minimal. It should not contain any redundancy"”"”even if such redundancy would make the definition much more intuitive.

- For everyone else ("random people on the street"), the goals are convenience and ease of understanding"”"”both of which are quite often facilitated by redundancy.

[RonPurewal]

As an example, "rectangle" would be defined by mathematicians as "a quadrilateral with at least three 90º angles". (Think about it for a second.) That's the bare minimum.
That definition is not at all convenient or intuitively accessible, though, so "people on the street" would instead opt for a definition containing some redundancy. E.g., "a quadrilateral with four 90º angles", or even "a quadrilateral with four 90º angles and two pairs of equal-length opposite sides".

[griffin.811]

You've been a huge help Ron! Thanks for the clarification.

[RonPurewal]

Sure.