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# M14-32

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Math Expert
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M14-32  [#permalink]

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15 Sep 2014, 23:54
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Difficulty:

85% (hard)

Question Stats:

45% (00:54) correct 55% (01:08) wrong based on 105 sessions

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If points $$A$$, $$B$$, $$C$$, and $$D$$ form a quadrilateral, is $$AC$$ longer than $$BD$$?

(1) $$\angle ABC \gt \angle BCD$$

(2) $$AB = BC = CD = DA$$

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M14-32  [#permalink]

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15 Sep 2014, 23:54
Official Solution:

If points A, B, C, and D form a quadrilateral, is AC longer than BD?

Statement (1) by itself is insufficient. Imagine a kite-shaped figure with an obtuse angle on top.

Statement (2) by itself is insufficient. S2 means that the quadrilateral is a rhombus.

Statements (1) and (2) combined are sufficient. S2 means that the quadrilateral is a rhombus, while S1 tells us which diagonal of the rhombus is longer.

Answer: C
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Re: M14-32  [#permalink]

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05 Dec 2014, 21:14
Why does the angle determine the length of the diagonal?
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Re: M14-32  [#permalink]

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06 Dec 2014, 05:37
rsamant wrote:
Why does the angle determine the length of the diagonal?

Draw the image:
Attachment:
Lines-and-Quadrilaterals-08.png
Does it helps?
>> !!!

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Re: M14-32  [#permalink]

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21 Jun 2015, 09:08
Bunuel

Could you please show a case in which AC is shorter or equal to BD according to statement 1?
Having hard time seeing (kite-shaped as suggested) with obtuse angle BCD being less than angle ABC...

Thank you!!
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Re: M14-32  [#permalink]

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06 Jan 2016, 12:17
me too :/ maybe someone wants to be really kind and draw it ....?
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Re: M14-32  [#permalink]

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12 Jan 2016, 04:35
3
in the figure, as per statement, the angles are different but you can see the lengths different.
>> !!!

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Re: M14-32  [#permalink]

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31 Jul 2016, 05:36
saran3129 wrote:
in the figure, as per statement, the angles are different but you can see the lengths different.

Hi Buneul,

Could you please elaborate on the explanation. If 4 sides are equal in a rhombus, wont it have equal diagonals?

Cheers
Balaji
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Re: M14-32  [#permalink]

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01 Aug 2016, 01:20
Balajikarthick1990 wrote:
saran3129 wrote:
in the figure, as per statement, the angles are different but you can see the lengths different.

Hi Buneul,

Could you please elaborate on the explanation. If 4 sides are equal in a rhombus, won`t it have equal diagonals?

Cheers
Balaji

Not necessarily. If the diagonals of a rhombus are equal, then the rhombus is a square. Check rhombus with unequal diagonals below:
>> !!!

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Re: M14-32  [#permalink]

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20 Mar 2017, 14:40
Bunuel, can you please explain the relationship between the measure of angles and the diagonals of a rhombus. In the figure above if angle ABC > angle BCD, how does that make the diagonal AC> diagonal BD?
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Re: M14-32  [#permalink]

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21 Mar 2017, 02:57
vtomar20 wrote:
Bunuel, can you please explain the relationship between the measure of angles and the diagonals of a rhombus. In the figure above if angle ABC > angle BCD, how does that make the diagonal AC> diagonal BD?

Important properties of a triangle.
The shortest side is always opposite the smallest angle.
The longest side is always opposite the largest angle.
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Re: M14-32  [#permalink]

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03 Jun 2017, 00:43
Hi Bunuel,

I have a doubt can't the first statement itself say that which diagonal is longer. If we are applying the property of triangle taking both the statements, can't it be taken individually.

Regards,
Arpan
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Re: M14-32  [#permalink]

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23 Jun 2017, 04:56
Please Bunuel
may you elaborate on statement 1? Even with kite shaped figure i see that the bigger angle is opposite the biggest diagonal ? What is this i cant see?
It must be something obvious but i posted since someone else had the same doubt before..
Thanks
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Re: M14-32  [#permalink]

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23 Jun 2017, 10:20
2
alexlovesgmat wrote:
Please Bunuel
may you elaborate on statement 1? Even with kite shaped figure i see that the bigger angle is opposite the biggest diagonal ? What is this i cant see?
It must be something obvious but i posted since someone else had the same doubt before..
Thanks

Consider the image below:
Attachment:
Untitled.png

In both figures $$\angle ABC \gt \angle BCD$$ but in the first one $$AC>BD$$ and in the second one $$AC<BD$$.

Hope it helps.
>> !!!

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Re: M14-32  [#permalink]

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24 Jun 2017, 09:18
Bunuel wrote:
Official Solution:

If points A, B, C, and D form a quadrilateral, is AC longer than BD?

Statement (1) by itself is insufficient. Imagine a kite-shaped figure with an obtuse angle on top.

Statement (2) by itself is insufficient. S2 means that the quadrilateral is a rhombus.

Statements (1) and (2) combined are sufficient. S2 means that the quadrilateral is a rhombus, while S1 tells us which diagonal of the rhombus is longer.

Answer: C

Bunuel

A rhombus with equal sides means that the diagonals are equal. Choice B should be sufficient alone.

Why do you consider it wrong?
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Joined: 02 Sep 2009
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Re: M14-32  [#permalink]

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24 Jun 2017, 09:36
Mo2men wrote:
Bunuel wrote:
Official Solution:

If points A, B, C, and D form a quadrilateral, is AC longer than BD?

Statement (1) by itself is insufficient. Imagine a kite-shaped figure with an obtuse angle on top.

Statement (2) by itself is insufficient. S2 means that the quadrilateral is a rhombus.

Statements (1) and (2) combined are sufficient. S2 means that the quadrilateral is a rhombus, while S1 tells us which diagonal of the rhombus is longer.

Answer: C

Bunuel

A rhombus with equal sides means that the diagonals are equal. Choice B should be sufficient alone.

Why do you consider it wrong?

Please read the whole thread: https://gmatclub.com/forum/m14-184027.html#p1717536

P.S. Rhombus by definition has equal sides.
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Re: M14-32  [#permalink]

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24 Jun 2017, 09:40
1
Mo2men wrote:
Bunuel wrote:
Official Solution:

If points A, B, C, and D form a quadrilateral, is AC longer than BD?

Statement (1) by itself is insufficient. Imagine a kite-shaped figure with an obtuse angle on top.

Statement (2) by itself is insufficient. S2 means that the quadrilateral is a rhombus.

Statements (1) and (2) combined are sufficient. S2 means that the quadrilateral is a rhombus, while S1 tells us which diagonal of the rhombus is longer.

Answer: C

Bunuel

A rhombus with equal sides means that the diagonals are equal. Choice B should be sufficient alone.

Why do you consider it wrong?

Hi

A Rhombus is a quadrilateral that has all 4 sides equal, but that does NOT necessarily mean that its diagonals will also be equal.
Bunuel has already explained it in this thread by the means of a diagram before, please check.

A Rhombus whose diagonals are made equal, then becomes a Square.
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Re: M14-32  [#permalink]

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07 Mar 2018, 22:54
can anyone please elaborate why statement A alone is not sufficient as we use property of triangle that larger angle will have opposite side larger which is conveyed by statement 1 and it straight away tell us AC>BD so answer must be A only because second statement tell us all sides equal then if its a square then we can say no and if its a rhombus then may be so insufficient please help he
#bunuel Bunuel
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Re: M14-32  [#permalink]

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08 Mar 2018, 00:46
1
rishabhmishra wrote:
can anyone please elaborate why statement A alone is not sufficient as we use property of triangle that larger angle will have opposite side larger which is conveyed by statement 1 and it straight away tell us AC>BD so answer must be A only because second statement tell us all sides equal then if its a square then we can say no and if its a rhombus then may be so insufficient please help he
#bunuel Bunuel

AC and BC are NOT in the same triangle so you cannot apply this property here. You can check the diagram above which shows this clearly: https://gmatclub.com/forum/m14-184027.html#p1875155
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Re: M14-32  [#permalink]

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08 Mar 2018, 03:32
Bunuel wrote:
rishabhmishra wrote:
can anyone please elaborate why statement A alone is not sufficient as we use property of triangle that larger angle will have opposite side larger which is conveyed by statement 1 and it straight away tell us AC>BD so answer must be A only because second statement tell us all sides equal then if its a square then we can say no and if its a rhombus then may be so insufficient please help he
#bunuel Bunuel

AC and BC are NOT in the same triangle so you cannot apply this property here. You can check the diagram above which shows this clearly: https://gmatclub.com/forum/m14-184027.html#p1875155

THANKS a lot sir
Re: M14-32 &nbs [#permalink] 08 Mar 2018, 03:32
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