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# In the figure above, P and Q are centers of two identical

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Manager
Joined: 20 Jun 2012
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Kudos [?]: 42 [0], given: 52

In the figure above, P and Q are centers of two identical [#permalink]

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25 Jun 2013, 05:15
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In the figure above, P and Q are centers of two identical circles. Is quadrilateral a square?

(1) The length of arc MN is 2pi.
(2) The area of one circle is 16 pi.

[Reveal] Spoiler:
Here NQ and QN are tangents to left circle, and we know any line from center of a circle to the point where tangent touches the circle will make a 90 degree angle with tangent.

Following this rule angle PNQ and PMQ are 90. and angle NPM and NQM must be equal to each other hence these are also 90. PN=PM=QN=QM=radius .. given. Hence this should be a square without considering any options .. what am I doing wrong ??
[Reveal] Spoiler: OA

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Re: In the figure above, P and Q are centers of two identical [#permalink]

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25 Jun 2013, 06:10
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In the figure above, P and Q are centers of two identical circles. Is quadrilateral a square?

(1) The length of arc MN is 2pi. Clearly insufficient.

(2) The area of one circle is 16 pi --> $$\pi{r^2}=16\pi$$ --> $$r=4$$. Not sufficient.

(1)+(2) From (2) the circumference of each circle is $$2\pi{r}=8\pi$$. So, we have that arc MN ($$2\pi$$) is 1/4th of the circumference which means that angles P and Q are 1/4*360=90 degrees. Since all 4 sides of the quadrilateral are equal and two sides are 90 degrees, then it must be a square. Sufficient.

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Kudos [?]: 1403 [2] , given: 197

Re: In the figure above, P and Q are centers of two identical [#permalink]

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25 Jun 2013, 14:09
2
KUDOS
stunn3r wrote:

In the figure above, P and Q are centers of two identical circles. Is quadrilateral a square?

(1) The length of arc MN is 2pi.
(2) The area of one circle is 16 pi.

[Reveal] Spoiler:
Here NQ and QN are tangents to left circle, and we know any line from center of a circle to the point where tangent touches the circle will make a 90 degree angle with tangent.

Following this rule angle PNQ and PMQ are 90. and angle NPM and NQM must be equal to each other hence these are also 90. PN=PM=QN=QM=radius .. given. Hence this should be a square without considering any options .. what am I doing wrong ??

hi,
this method of yours will not hold in all cases.
let us suppose left circle passes through centre of right circle...in that case QM AND QN will not be tangent then you cant make it 90 on the basis of that.
might there be other method to prove that a square..but your method will not hold true....
KUDOS if it helped.
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Math Expert
Joined: 02 Sep 2009
Posts: 39050
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Kudos [?]: 106532 [0], given: 11627

Re: In the figure above, P and Q are centers of two identical [#permalink]

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25 Jun 2013, 14:14
shaileshmishra wrote:
stunn3r wrote:

In the figure above, P and Q are centers of two identical circles. Is quadrilateral a square?

(1) The length of arc MN is 2pi.
(2) The area of one circle is 16 pi.

[Reveal] Spoiler:
Here NQ and QN are tangents to left circle, and we know any line from center of a circle to the point where tangent touches the circle will make a 90 degree angle with tangent.

Following this rule angle PNQ and PMQ are 90. and angle NPM and NQM must be equal to each other hence these are also 90. PN=PM=QN=QM=radius .. given. Hence this should be a square without considering any options .. what am I doing wrong ??

hi,
this method of yours will not hold in all cases.
let us suppose left circle passes through centre of right circle...in that case QM AND QN will not be tangent then you cant make it 90 on the basis of that.
might there be other method to prove that a square..but your method will not hold true....
KUDOS if it helped.

Where in the solution did you see that 90 degrees were deduced based on tangency?
_________________
Director
Joined: 14 Dec 2012
Posts: 832
Location: India
Concentration: General Management, Operations
GMAT 1: 700 Q50 V34
GPA: 3.6
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Kudos [?]: 1403 [0], given: 197

Re: In the figure above, P and Q are centers of two identical [#permalink]

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25 Jun 2013, 14:19
Bunuel wrote:
shaileshmishra wrote:
stunn3r wrote:

In the figure above, P and Q are centers of two identical circles. Is quadrilateral a square?

(1) The length of arc MN is 2pi.
(2) The area of one circle is 16 pi.

[Reveal] Spoiler:
Here NQ and QN are tangents to left circle, and we know any line from center of a circle to the point where tangent touches the circle will make a 90 degree angle with tangent.

Following this rule angle PNQ and PMQ are 90. and angle NPM and NQM must be equal to each other hence these are also 90. PN=PM=QN=QM=radius .. given. Hence this should be a square without considering any options .. what am I doing wrong ??

hi,
this method of yours will not hold in all cases.
let us suppose left circle passes through centre of right circle...in that case QM AND QN will not be tangent then you cant make it 90 on the basis of that.
might there be other method to prove that a square..but your method will not hold true....
KUDOS if it helped.

Where in the solution did you see that 90 degrees were deduced based on tangency?

hi bunuel,
actually i repled to the spoiler of the author (stunn3r).
if possible please refer to the spoiler.
SKM
_________________

When you want to succeed as bad as you want to breathe ...then you will be successfull....

GIVE VALUE TO OFFICIAL QUESTIONS...

learn AWA writing techniques while watching video : http://www.gmatprepnow.com/module/gmat-analytical-writing-assessment

Math Expert
Joined: 02 Sep 2009
Posts: 39050
Followers: 7753

Kudos [?]: 106532 [0], given: 11627

Re: In the figure above, P and Q are centers of two identical [#permalink]

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25 Jun 2013, 14:22
shaileshmishra wrote:
Bunuel wrote:
shaileshmishra wrote:

hi,
this method of yours will not hold in all cases.
let us suppose left circle passes through centre of right circle...in that case QM AND QN will not be tangent then you cant make it 90 on the basis of that.
might there be other method to prove that a square..but your method will not hold true....
KUDOS if it helped.

Where in the solution did you see that 90 degrees were deduced based on tangency?

hi bunuel,
actually i repled to the spoiler of the author (stunn3r).
if possible please refer to the spoiler.
SKM

My bad. Missed the text under the spoiler.
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GMAT 1: 710 Q51 V25
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Kudos [?]: 42 [0], given: 52

Re: In the figure above, P and Q are centers of two identical [#permalink]

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25 Jun 2013, 15:40
shaileshmishra wrote:
stunn3r wrote:

In the figure above, P and Q are centers of two identical circles. Is quadrilateral a square?

(1) The length of arc MN is 2pi.
(2) The area of one circle is 16 pi.

[Reveal] Spoiler:
Here NQ and QN are tangents to left circle, and we know any line from center of a circle to the point where tangent touches the circle will make a 90 degree angle with tangent.

Following this rule angle PNQ and PMQ are 90. and angle NPM and NQM must be equal to each other hence these are also 90. PN=PM=QN=QM=radius .. given. Hence this should be a square without considering any options .. what am I doing wrong ??

hi,
this method of yours will not hold in all cases.
let us suppose left circle passes through centre of right circle...in that case QM AND QN will not be tangent then you cant make it 90 on the basis of that.
might there be other method to prove that a square..but your method will not hold true....
KUDOS if it helped.

That helped .. THank You Shailesh .. and forgive BUNU, he misses things sometimes ..
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Joined: 29 May 2013
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Kudos [?]: 1 [0], given: 3

Re: In the figure above, P and Q are centers of two identical [#permalink]

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10 Jul 2013, 02:40
Hi Brunel,

Wont angle PMQ and PNQ be equal to 90 degrees considering the tanget properties.

Also, if we divide the quadilateral by drawing the line PQ, angle MPQ=MQP = 45 degrees as angle PMQ= 90 degree. Similarly, angle QPN will be 45 degrees, so angle P= angle MPQ+ angle QPN = 45+45=90

Please correct me if wrong .
Math Expert
Joined: 02 Sep 2009
Posts: 39050
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Kudos [?]: 106532 [0], given: 11627

Re: In the figure above, P and Q are centers of two identical [#permalink]

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10 Jul 2013, 02:45
Kriti2013 wrote:
Hi Brunel,

Wont angle PMQ and PNQ be equal to 90 degrees considering the tanget properties.

Also, if we divide the quadilateral by drawing the line PQ, angle MPQ=MQP = 45 degrees as angle PMQ= 90 degree. Similarly, angle QPN will be 45 degrees, so angle P= angle MPQ+ angle QPN = 45+45=90

Please correct me if wrong .

We can derive all these when we combine the statements, but not from the beginning, since we don't know whether radii are the tangents.

Hope it's clear.
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Re: In the figure above, P and Q are centers of two identical [#permalink]

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31 Jul 2014, 09:33
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Re: In the figure above, P and Q are centers of two identical [#permalink]

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10 Sep 2015, 08:22
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Re: In the figure above, P and Q are centers of two identical [#permalink]

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29 Nov 2016, 13:41
Hello from the GMAT Club BumpBot!

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Re: In the figure above, P and Q are centers of two identical   [#permalink] 29 Nov 2016, 13:41
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