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In quadrilateral ABCD, sides AB and BC each have length √2,

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In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 04 Sep 2012, 04:06
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In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.
[Reveal] Spoiler: OA

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Last edited by Bunuel on 30 Mar 2014, 10:14, edited 1 time in total.
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 10 Sep 2012, 10:57
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i guess with both, statements 1 and 2 together, you are able to determine the area of the given quadrilateral.

please refer to the image attached..

Attachment:
quadABCD.jpg
quadABCD.jpg [ 19.47 KiB | Viewed 5119 times ]


you can see that once you have the 2nd condition, you can find the diagonal AC of the quad.
and when you calculate the length of the diagonal, you end up getting an equilateral triangle ADC on the other side.
So, 1 you have a right triangle and 2 you have an equilateral triangle, the areas of both of which can be calculated.

Thus, I would go with option C.

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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 30 Mar 2014, 10:16
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jlgdr wrote:
imhimanshu wrote:
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.

Please post your reasoning.


I'm going with A here. Please provide OA ok?

Here's my reasoning

Statement 1 we have that AD which is the diagonal of the right triangle formed in the polygon is 2 therefore knowing two sides and one angle is sufficient to know the measures of the right triangle but thing is we don't know about BC is this side perpendicular to AB or not? Insufficient

Statement 2 only tells us that the angle in ABC is 90 degrees but still no way to find the area of the other right triangle.

Now with both statements together we know the area of the right triangle and we know that AB is perpendicular to BC hence the area of the given square is 2 which added to the area of the right triangle can provide the total area of the figure

Answer is thus C

Please let me know if this is sound reasoning
Cheers
J


The correct answer is E:
Attachment:
quadrophenia-3.jpg
quadrophenia-3.jpg [ 9.69 KiB | Viewed 1753 times ]

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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 04 Sep 2012, 04:43
Statement (1): We know all 4 sides. But we don't know any angle. There is more than one way to draw this quadrilateral. --> not sufficient
Statement (2): Again, there's one info too few. We would need one more angle or the fourth side. There is more than one way to draw this quadrilateral --> not sufficient

Both statements together are sufficient. We can draw two triangles ABC and CDA. ABC is an isosceles rectangular triangle. What's important here is that CD and DA are longer than AB and BC. If this would not be the case, we could draw two quadrilaterals, one with angle CDA being smaler than 180 degrees, the other one with angle CDA being greater than 180 degrees.

We could mirror/flip triangle CDA anyways, but this way, the angle ABC would not longer be 90 degrees, but 270 degrees (the 90 degrees would then be "outside".
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Challenge Problem from Manhattan GMAT [#permalink] New post 04 Sep 2012, 13:28
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.


I was able to solve the datasufficiency question using the 1 and 2 but the OA is E.My Approach is given below. GMAT club members kindly check and let me know whether there is anything wrong with my approach.


Area of quadrilateral=1\2 *d1(p1+p2) d1=lenght of one digaonal;p1=lenght of perpendicular from one vertex.p2=lenght of perpendicular from opp vertex to the diagonal.

1.Using 1. In traingle ADC of quadrilateral ABCD AD=DC. Draw perpendicular say DE from vertex D onto diagonal AC of quadrilateral.Now traingles ADE and CDE are congruent traingles AAS congruency(<DAE=<DCA ;<DEA=<DEC=90 ;AD=DC), so AE=EC.But we cant calculate length of diagonal AC from the info, hence it is insufficient.

2.Using this info we can calculate the length of diagonal AC from pythagoras theorem.Lenghth of other perpendicular drawn from vertex B on AC say BF can be found out using the same procedure as above ie by AAS congruency.But we cant calculate lenght of other perpendicular drawn from vertex D with this infro.Hence this is aslo insufficient.

Using (1)+(2) we get Length of diagonal and lenght of perpendicular DE and BF (E and F will be the same point as from AAS congruency of two traingles we get AE+EC =AC and AF=FC=AC).Hence both together are sufficeint to answer the problem.

Let me know your views on this problem.
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 05 Sep 2012, 13:47
I believe it's E.

Statement 1 is not sufficient because we don't know if angle ABD is acute or obtuse.

Statement 2 is not sufficient because of aforementioned reasons.

Even with both statements we still have nothing about the angle ABD.
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 07 Sep 2012, 03:44
if we know four sides of the quadrilateral we can find its area...

s=a+b+c+d/4

area =sqrt of s*(s-a)*(s-b)*(s-c)*(s-d) ....

so answer A ...

Am i missing something.....
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 11 Sep 2012, 11:42
imhimanshu wrote:
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.

Please post your reasoning.


I go with answer 'D' as both the sentences are sufficient to give the answer.
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 10 Oct 2012, 04:21
imhimanshu wrote:
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.

Please post your reasoning.



From stem
AB = BC = 2^1/2
CD = 2

1) AD = 2 ---> This can form a KITE if we knew the angles but since there is no info about the angles included -->Insufficient
2) One of the triangle is right angle triangle but no info about the shape of other triangle -->Insufficient.
1+2) If we combine both we get KITE ( a combination of 2 Isosceles Right angle triangles) ---> Sufficient

Answer C
Note :- Not able to include the diagram, you have to imagine.

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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 10 Oct 2012, 09:59
imhimanshu wrote:
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.

Please post your reasoning.


To the best of my knowledge, on the GMAT only convex polygons are considered, except maybe the cases when a drawing is supplied.
This is just another example when Manhattan is going overboard.

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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 30 Mar 2014, 08:17
imhimanshu wrote:
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.

Please post your reasoning.


I'm going with C here. Please provide OA ok?

Here's my reasoning

Statement 1 we have that AD which is the diagonal of the right triangle formed in the polygon is 2 therefore knowing two sides and one angle is sufficient to know the measures of the right triangle but thing is we don't know about BC is this side perpendicular to AB or not? Insufficient

Statement 2 only tells us that the angle in ABC is 90 degrees but still no way to find the area of the other right triangle.

Now with both statements together we know the area of the right triangle and we know that AB is perpendicular to BC hence the area of the given square is 2 which added to the area of the right triangle can provide the total area of the figure

Answer is thus C

Please let me know if this is sound reasoning
Cheers
J

Last edited by jlgdr on 07 Apr 2014, 04:51, edited 1 time in total.
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 17 Apr 2014, 10:36
Bunuel wrote:
jlgdr wrote:
imhimanshu wrote:
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.

Please post your reasoning.


I'm going with A here. Please provide OA ok?

Here's my reasoning

Statement 1 we have that AD which is the diagonal of the right triangle formed in the polygon is 2 therefore knowing two sides and one angle is sufficient to know the measures of the right triangle but thing is we don't know about BC is this side perpendicular to AB or not? Insufficient

Statement 2 only tells us that the angle in ABC is 90 degrees but still no way to find the area of the other right triangle.

Now with both statements together we know the area of the right triangle and we know that AB is perpendicular to BC hence the area of the given square is 2 which added to the area of the right triangle can provide the total area of the figure

Answer is thus C

Please let me know if this is sound reasoning
Cheers
J


The correct answer is E:
Attachment:
quadrophenia-3.jpg


Hi Bunnel,

When I use st1 and st2 I get one 45-45-90 triangle. and another one an equilateral triangle with site 2. I can get area of quadrilateral by adding both the areas. I choose C but official ans is E. Please clarify.

Thanks
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 18 Apr 2014, 01:45
Expert's post
PathFinder007 wrote:
Bunuel wrote:
jlgdr wrote:
In quadrilateral ABCD, sides AB and BC each have length √2, while side CD has length 2. What is the area of quadrilateral ABCD?

(1) The length of side AD is 2.
(2) The angle between side AB and side BC is 90°.


I'm going with A here. Please provide OA ok?

Here's my reasoning

Statement 1 we have that AD which is the diagonal of the right triangle formed in the polygon is 2 therefore knowing two sides and one angle is sufficient to know the measures of the right triangle but thing is we don't know about BC is this side perpendicular to AB or not? Insufficient

Statement 2 only tells us that the angle in ABC is 90 degrees but still no way to find the area of the other right triangle.

Now with both statements together we know the area of the right triangle and we know that AB is perpendicular to BC hence the area of the given square is 2 which added to the area of the right triangle can provide the total area of the figure

Answer is thus C

Please let me know if this is sound reasoning
Cheers
J


The correct answer is E:
Attachment:
quadrophenia-3.jpg


Hi Bunnel,

When I use st1 and st2 I get one 45-45-90 triangle. and another one an equilateral triangle with site 2. I can get area of quadrilateral by adding both the areas. I choose C but official ans is E. Please clarify.

Thanks


Not sure how to clarify better than the diagram:
Image

As you can see both those figure satisfy all the conditions and have different areas. Thus the answer must be E.

Hope it helps.

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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 19 Apr 2014, 00:30
Thanks for clarification. But i think second figure is not a convex polygon(because interior ABC angle is 270 and not less than 180). And in GMAT I guess we are concerned about convex polygon only?
And if we agree that second figure is valid, then can we say ans is C? Also i feel is it really possible to draw such diagram with precision in exam :)? Any alternative way to build the thought process for exam?

Thanks.
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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 21 Apr 2014, 04:09
Expert's post
manojISB wrote:
Thanks for clarification. But i think second figure is not a convex polygon(because interior ABC angle is 270 and not less than 180). And in GMAT I guess we are concerned about convex polygon only?
And if we agree that second figure is valid, then can we say ans is C? Also i feel is it really possible to draw such diagram with precision in exam :)? Any alternative way to build the thought process for exam?

Thanks.
Manoj Parashar


Yes, OG defines a quadrilateral as a polygon with four sides. Next, it says that the term "polygon" will be used to mean a convex polygon, that is, a polygon in which each interior angle has a measure of less than 180°.

Hence this question violates GMAT definition of a quadrilateral, which makes the question/solution presented by MGMAT flawed.

I wouldn't worry about this question at all.

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Re: In quadrilateral ABCD, sides AB and BC each have length √2, [#permalink] New post 29 Jun 2014, 10:09
Is there a set of details that we must know in order to find the area of a quadrilateral?
Re: In quadrilateral ABCD, sides AB and BC each have length √2,   [#permalink] 29 Jun 2014, 10:09
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