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# R is a convex polygon. Does R have at least 8 sides? (1)

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VP
Joined: 21 Jul 2006
Posts: 1490
R is a convex polygon. Does R have at least 8 sides? (1) [#permalink]

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14 Aug 2008, 06:41
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R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.

(2) None of the interior angles of R are less than 60 degrees.

Thanks

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Director
Joined: 14 Aug 2007
Posts: 704

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14 Aug 2008, 07:14
A.

1) consider an 8 sides polygon, the sum of angles= 6*180 = 1080
Only 3 angles can be greater than 80 and the polygon is convex polygon so each angle must be < 180
let the measure of 3 angles each be 170
170* 3 = 510
sum of remaining angles = 1080 - 510 = 570

each of remaining angles= 570/3 = 190..which cant be true...

thus the polygon cant have atleast 8 sides.

suff

2) This can be true for any polygon, e.g. rectangle, hexagon or 9 sided polygon as well.

Thus insuff
Manager
Joined: 27 May 2008
Posts: 139

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14 Aug 2008, 09:23
tarek99 wrote:
R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.

(2) None of the interior angles of R are less than 60 degrees.

Thanks

A very very good question.

The sum total of an 8 sided convex polygon = 1080
If, three of the interior angles have a degree measure of more than 80, the rest of the 5 angles will have a degree measure of less than 80.

In a convex polygon none of the angle can be greater than 180.

considering the worst case scenario:

The 3 angles greater than 80, can at max have a degree measure of 179.
And the rest of the five angles, can at max have a degree measure of 79.

179*3 + 79*5 < 1080

The second is too broad of an option to give any kind of an answer.
SVP
Joined: 07 Nov 2007
Posts: 1760
Location: New York

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14 Aug 2008, 09:33
tarek99 wrote:
R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.

(2) None of the interior angles of R are less than 60 degrees.

Thanks

SUM OF THE INTERNAL ANGLES FOR Convex POLYGON with n sides = (n-2)*180 = 6*180

1)Exactly 3 of the interior angles of R are greater than 80 degrees.
important for onvex polygon .. all angles must be <180 degrees.
these three angles must be between netween 80 and 180
assume that 3 angles are ~ 180 (max possible angles.
3*180 + A1+A2+...An= 6*180 for atleast 8 sides n must be >=5
A1+A2+... An ~ 3*180
A1.. An must be <=80
assume that all are 80
n= (3*180/80)=54/8=6.7 ~7
So n must be atleast 7 .. means.. no of angles >10
Suffcieint.
2)None of the interior angles of R are less than 60 degrees.
Clearly this is infufficient... possible.. n>8 and n<8
rectangle. -- all are 90 >60
octogon -- possible.. all angles must be equal to 120.

Insufficient.

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VP
Joined: 21 Jul 2006
Posts: 1490

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14 Aug 2008, 10:15
rahulgoyal1986 wrote:
tarek99 wrote:
R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.

(2) None of the interior angles of R are less than 60 degrees.

Thanks

A very very good question.

The sum total of an 8 sided convex polygon = 1080
If, three of the interior angles have a degree measure of more than 80, the rest of the 5 angles will have a degree measure of less than 80.

In a convex polygon none of the angle can be greater than 180.

considering the worst case scenario:

The 3 angles greater than 80, can at max have a degree measure of 179.
And the rest of the five angles, can at max have a degree measure of 79.

179*3 + 79*5 < 1080

The second is too broad of an option to give any kind of an answer.

Thanks everyone for your great input! The OA is A, but I have a question regarding your approach. You said that when 3 angles are over 80 degrees, then the remaining 5 angles must be less than 80 degrees? Would you please explain why the remaining 5 angles must be less than 80 degrees?
VP
Joined: 21 Jul 2006
Posts: 1490

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14 Aug 2008, 10:26
question guys. Does a convex polygon mean that each angle is less than 180 degrees or does it mean that each angle is less than or equal to 180 degrees?
thanks
SVP
Joined: 30 Apr 2008
Posts: 1850
Location: Oklahoma City
Schools: Hard Knocks

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14 Aug 2008, 10:28
An angle of 180 degrees isn't really much of an angle in geometry. Technically speaking, it's possible, but with drawing a polygon, show me an enclosed shape with 10 angles of 180 degrees each

If you trust Wikipedia:

http://en.wikipedia.org/wiki/Convex_polygon

It says:

Convex = Angle < 180 degrees
Concave = Angle > 180 degrees

Angle = 180 degrees = straight line.

EDIT:

I think the best definition should be:
Convex = All Angles <= 180 degrees
Concave = All Angles >= 180 degrees

A circle is not a polygon but it "could" satisfy the definition of Covnex = Angle < 180 degrees. If you thought about each pixel on your monitor being a point on the circle as it is displayed, and measured the angles between them, there would technically not be an angle that measures 180 degrees.

But the midpoint of a side of a square does technically measure 180 degrees.

This is a point that doesn't really matter in the end for GMAT. Does a circle even have "angles"?

tarek99 wrote:
question guys. Does a convex polygon mean that each angle is less than 180 degrees or does it mean that each angle is less than or equal to 180 degrees?
thanks

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SVP
Joined: 07 Nov 2007
Posts: 1760
Location: New York

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14 Aug 2008, 11:02
tarek99 wrote:
question guys. Does a convex polygon mean that each angle is less than 180 degrees or does it mean that each angle is less than or equal to 180 degrees?
thanks

Convex polygon means.. all angles must be < 180

If you try to draw any polygon with all angles equal to 180 degrees you will get a straight line

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_________________

Smiling wins more friends than frowning

Manager
Joined: 27 May 2008
Posts: 139

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14 Aug 2008, 20:47
tarek99 wrote:
rahulgoyal1986 wrote:
tarek99 wrote:
R is a convex polygon. Does R have at least 8 sides?

(1) Exactly 3 of the interior angles of R are greater than 80 degrees.

(2) None of the interior angles of R are less than 60 degrees.

Thanks

A very very good question.

The sum total of an 8 sided convex polygon = 1080
If, three of the interior angles have a degree measure of more than 80, the rest of the 5 angles will have a degree measure of less than 80.

In a convex polygon none of the angle can be greater than 180.

considering the worst case scenario:

The 3 angles greater than 80, can at max have a degree measure of 179.
And the rest of the five angles, can at max have a degree measure of 79.

179*3 + 79*5 < 1080

The second is too broad of an option to give any kind of an answer.

Thanks everyone for your great input! The OA is A, but I have a question regarding your approach. You said that when 3 angles are over 80 degrees, then the remaining 5 angles must be less than 80 degrees? Would you please explain why the remaining 5 angles must be less than 80 degrees?

I meant, as per the question, if three out of eight angles of an octagon are above 80 degrees, the rest 5 will be less than 80. Had the rest of the angles been greater than 80 thn the question would have said more than 3 angles of the polygon are more thn 80.

Hope above helps. Feel free to quote again
VP
Joined: 21 Jul 2006
Posts: 1490

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15 Aug 2008, 03:22
so, if 3 angles are suppose to be greater than 80 degrees, shouldn't that mean that the remaining 5 angles must be less than or equal to 80? I was just wondering why you rather specifically went for the less than 80 degrees.
Manager
Joined: 27 May 2008
Posts: 139

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15 Aug 2008, 20:46
tarek99 wrote:
so, if 3 angles are suppose to be greater than 80 degrees, shouldn't that mean that the remaining 5 angles must be less than or equal to 80? I was just wondering why you rather specifically went for the less than 80 degrees.

Well Well Well. Mr. Tarek u're so very right.

Thanks for pointing it out. Cheers!!

--== Message from GMAT Club Team ==--

This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

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Re: DS: Geometry   [#permalink] 15 Aug 2008, 20:46
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