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29 Dec 2010, 08:08
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ABCD is a quadrilateral. A rhombus is a quadrilateral whose sides are all congruent. BCEF is a rhombus and shares a common side with the quadrilateral ABCD. The area of which one is greater: ABCD or BCEF ?

(1) ABCD is a square.
(2) BCEF is not a square.
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Re: Area of which is greater ?? [#permalink]

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29 Dec 2010, 10:30
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Let's do it step by step.

1. How are ABCD and BCEF connected? There is only a common side BC. All other tree sides of ABCD can be of any size and the area of ABCD can be (0, inf). BCEF is a rhombus and its area can be (0, $$BC^2$$] ($$BC^2$$ included).

2. Statement 1. In other words, area of ABCD is $$BC^2$$ that can be greater or equal to the area of BCEF (0, $$BC^2$$]

3. Statement 2. So, area of BCEF can't be $$BC^2$$ and we have (0, $$BC^2$$) possible range for the area. it's still not enough as the area ABCD can be (0, inf)

4. Statements 1&2. The area of ABCD is $$BC^2$$ and the area of BCEF can be (0, $$BC^2$$) or always lesser than the area of ABCD. Sufficient.

C
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01 Apr 2012, 05:04
Hi

Iam new to GMAT club and was going through this problem.I think A should be the answer to this problem.Logic given below.

1.Statement 1 says ABCD is a square. Now for all quadrilaterals with given perimeter square has the greatest area .Perimeter of square ABCD and BCEF =4a where a=length of one side.(Square and rhombus share a common side ,so lenght of side of Rhombus is also a).So applying the above property area of square greater than rhombus.

2.) Statement 2 says that BCEF is not a square, doesnt provide any useful infromation.Hence clearly not sufficient.

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02 Apr 2012, 22:15
burp wrote:
ABCD is a quadrilateral. A rhombus is a quadrilateral whose sides are all congruent. BCEF is a rhombus and shares a common side with the quadrilateral ABCD. The area of which one is greater: ABCD or BCEF ?

(1) ABCD is a square.
(2) BCEF is not a square.

Hi All,

I am having difficulty in understanding the proble,

1. ABCD is a square....As per defination Rhombus also has the same side. I am confused on how to tell the diagonal lengths with the help of sides of a rhombus. I think this is not sufficient.

2. BCEF not a square, no info about ABCD..hence not sufficient.

Together..we know ABCD is a square and BECF is not, but it is still rhombus...hence it is not sufficient together too...so E

Can any expert help us understand this
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04 Jul 2012, 06:48
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harshavmrg wrote:
burp wrote:
ABCD is a quadrilateral. A rhombus is a quadrilateral whose sides are all congruent. BCEF is a rhombus and shares a common side with the quadrilateral ABCD. The area of which one is greater: ABCD or BCEF ?

(1) ABCD is a square.
(2) BCEF is not a square.

Hi All,

I am having difficulty in understanding the proble,

1. ABCD is a square....As per defination Rhombus also has the same side. I am confused on how to tell the diagonal lengths with the help of sides of a rhombus. I think this is not sufficient.

2. BCEF not a square, no info about ABCD..hence not sufficient.

Together..we know ABCD is a square and BECF is not, but it is still rhombus...hence it is not sufficient together too...so E

Can any expert help us understand this

Dear Harsh,

Q. is : ABCD is a quadrilateral. A rhombus is a quadrilateral whose sides are all congruent. BCEF is a rhombus and shares a common side with the quadrilateral ABCD. The area of which one is greater: ABCD or BCEF ?

(1) ABCD is a square.
(2) BCEF is not a square.

This question can be very easily solved with a property of a quadilateral:-
A square has a larger area than any other quadrilateral with the same perimeter
Now for (1) ABCD is a square, but we know nothing about BCEF. Another property is
If the diagonals of a rhombus are equal, then that rhombus must be a square. Now as we see,
we only know that BCEF is a Rhombus, we know nothing about the angle. Hence (1) insufficient.
(2) Insufficient
(1) + (2), here we get that BCEF is not a square, hence clearly Area ABCD > Area BCEF Sufficient
Hence C
Hope I am clear

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05 Jul 2012, 13:52
+1 C

The square, given a perimeter, is the quadrilateral with the greater area.
A rombus can be a square also.
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ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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04 Aug 2013, 09:18
ABCD is a quadrilateral. BCEF is a rhombus and shares a common side with the quadrilateral ABCD. The area of which one is greater ABCD or BCEF ?

(1) ABCD is a square.
(2) BCEF is not a square.

Solution:
From (1),
ABCD is a square. So its area= BC × BC
And we know from the question that BCEF is a rhombus which shares a side with ABCD,
so the area of the rhombus = BC × FJ (from diagram).
[The area of rhombus = area of two equal triangles
inside it = ½ BC× FJ + ½ BC× FJ = BC× FJ ]

Here all sides of the two quadrilaterals are equals for sharing one same side.

But a rhombus has a less altitude than a square has because of its angles.
So clearly BC>FJ,
Consequently area of ABCD > area of BCEF

Statement (1) alone is sufficient.

Statement (2) is not necessary at all, because from the question we already know BCEF is a rhombus. Its not sufficient at all.

From my point of view the answer is (A), but in the Nova’s data sufficiency prep course book (Author: jeff kolby) i find the answer is (C).

What do you people think? Share your opinion…
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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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04 Aug 2013, 09:49
a square is a rhombus too ( vice a versa is not true)
unless we know that rhombus is not square, we cannot tell area is greater or equal.
So if it is square then area is equal but if not square then square > rhombus.

So C

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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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04 Aug 2013, 12:58
rhombus has maximum area when it is a square otherwise in all other cases the area of the rhombus will be less than the square with same side length.

as square comes under category of a RHOMBUS so we need to consider that case also

statement 1:
ABCD is a square===>in this situation there are 2 cases
case 1:when rhombus is a square ===>in this case the area of both ABCD and CDEF will be same.
case 2:when rhombus is not a square===.in this case area of ABCD will be MORE THAN CDEF.
hence insufficient

statement 2:
CDEF is not a square
this alone doesnt gives any information about ABCD.hence insufficient.

combining :
now we know CDEF is not a square hence area ABCD will be greater than CDEF.
HENCE C

hope this helps
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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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04 Aug 2013, 14:03
AMITAGARWAL2 wrote:
a square is a rhombus too ( vice a versa is not true)
unless we know that rhombus is not square, we cannot tell area is greater or equal.
So if it is square then area is equal but if not square then square > rhombus.

So C

how can square be a rhombus.
All the angles are 90 degree in square but not even one in the rhombus !!!
we know it clearly...... with equal sides as square, a rhombus must have less altitude than that of the square due to the angles...
can you explain it?
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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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05 Aug 2013, 11:42
Asifpirlo wrote:
AMITAGARWAL2 wrote:
a square is a rhombus too ( vice a versa is not true)
unless we know that rhombus is not square, we cannot tell area is greater or equal.
So if it is square then area is equal but if not square then square > rhombus.

So C

how can square be a rhombus.
All the angles are 90 degree in square but not even one in the rhombus !!!
we know it clearly...... with equal sides as square, a rhombus must have less altitude than that of the square due to the angles...
can you explain it?

A rhombus is a quadrilateral with all 4 sides equal in length. A square is a type of rhombus just as a rhombus is a type of quadrilateral.

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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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05 Aug 2013, 11:45
Observer wrote:
Asifpirlo wrote:
AMITAGARWAL2 wrote:
a square is a rhombus too ( vice a versa is not true)
unless we know that rhombus is not square, we cannot tell area is greater or equal.
So if it is square then area is equal but if not square then square > rhombus.

So C

how can square be a rhombus.
All the angles are 90 degree in square but not even one in the rhombus !!!
we know it clearly...... with equal sides as square, a rhombus must have less altitude than that of the square due to the angles...
can you explain it?

A rhombus is a quadrilateral with all 4 sides equal in length. A square is a type of rhombus just as a rhombus is a type of quadrilateral.

yes its a proven theory that a square is a rhombus too despite the differences in angles......
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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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05 Aug 2013, 13:51
Asifpirlo wrote:
yes its a proven theory that a square is a rhombus too despite the differences in angles......

Sorry misunderstood your question. I don't know if that diagram you posted is your own or is from the book you got the problem from, but it's misleading. Without B you do not know whether BCEF is a square or not. The diagram suggests we know that BCEF is a rhombus and not a square from the stem but only B gives that information.

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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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05 Aug 2013, 14:12
Observer wrote:
Asifpirlo wrote:
yes its a proven theory that a square is a rhombus too despite the differences in angles......

Sorry misunderstood your question. I don't know if that diagram you posted is your own or is from the book you got the problem from, but it's misleading. Without B you do not know whether BCEF is a square or not. The diagram suggests we know that BCEF is a rhombus and not a square from the stem but only B gives that information.

the question tells BCEF is a rhombus and accordint to it i draw the diagram...
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Re: ABCD is a quadrilateral. BCEF is a rhombus and shares a comm [#permalink]

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09 Dec 2013, 19:03
ABCD is a quadrilateral. BCEF is a rhombus and shares a common side with the quadrilateral ABCD. The area of which one is greater ABCD or BCEF ?

(1) ABCD is a square.

Tells us that ABCD is a square and that BCEF shares side measurements with ABCD. A rhombus will have a smaller than a square with the same side lengths. However, BCEF could be a square as well because a square is also a rhombus. We cannot be 100% sure. Not Sufficient.

(2) BCEF is not a square.

Tells us nothing about ABCD and very little about BCEF. Not Sufficient.

1+2) Tells us that ABCD is a square and that BCEF has the same side measurements but is not a square. Therefore, the area of ABCD is greater than the area of BCEF. SUFFICIENT

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10 Nov 2015, 11:29
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

ABCD is a quadrilateral. A rhombus is a quadrilateral whose sides are all congruent. BCEF is a rhombus and shares a common side with the quadrilateral ABCD. The area of which one is greater: ABCD or BCEF ?

(1) ABCD is a square.
(2) BCEF is not a square.

There are 2 variables (length of one side, diagonal) in a rhombus, 5variables (as 2 triangles are side by side) and 1 equation is given as they share one side in a quadrilateral. We are only given 2 equations from the conditions, so there is high chance (E) will be our answer.
Looking at the conditions together, ABCD becomes a square, and for BCEF, one side is shared so the area is greatest if it is a square.
Hence, ABCD's area>BCEF's area. The conditions are sufficient, and the answer becomes (C).

For cases where we need 3 more equation, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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