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

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R is a convex polygon. Does R have at least 8 sides?  [#permalink]

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Updated on: 22 Apr 2014, 02:48
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95% (hard)

Question Stats:

25% (02:12) correct 75% (02:10) wrong based on 221 sessions

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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.

Originally posted by kevincan on 13 Aug 2006, 15:44.
Last edited by Bunuel on 22 Apr 2014, 02:48, edited 1 time in total.
Edited the question and added the OA.
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Re: R is a convex polygon. Does R have at least 8 sides? (1)  [#permalink]

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22 Apr 2014, 03:45
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pretzel wrote:
IMO is E.

A convex polygon with 4 sides can still have angles greater than 80 and less than 60. Experts advice on this?

No, the correct answer is A.

R is a convex polygon. Does R have at least 8 sides?

The Sum of Interior Angles of a polygon is $$180(n-2)$$ degrees, where $$n$$ is the number of sides (so is the number of angles). So, the greater the number of sides the greater is the sum of the angles.

For 8 sided polygon the sum of the angles is $$180(n-2)=180*6=1080$$ degrees. The question basically asks whether the sum of the angles of the polygon is more than or equal to 1080 degrees.

(1) Exactly 3 of the interior angles of R are greater than 80 degrees. This implies that each of the remaining angles must be less than or equal to 80 degrees.

Assume that the polygon IS 8-sided. In this case, the sum of those 3 angles would be $$3*80<(sum \ of \ the \ given \ 3 \ angles)<3*180$$ --> $$240<(sum \ of \ the \ given \ 3 \ angles)<540$$.

Thus the sum of the reaming 5 angles must be $$1080-540<(sum \ of \ the \ remaining \ 5 \ angles)<1080-240$$ --> $$540<(sum \ of \ the \ remaining \ 5 \ angles)<840$$. But the sum of the reaming 5 angles must be less than or equal to 5*80=400, and not greater than 540 degrees. Therefore the assumption that the polygon could be 8-sided was wrong.

If the polygon cannot be 8-sided, then it cannot be more sided too. So, R must have less than 8 sides. Sufficient.

(2) None of the interior angles of R are less than 60 degrees. Not sufficient: consider equilateral triangle (all angles 60 degrees) and regular octagon (all angles 1080/8=135 degrees).

Hope it's clear.
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24 Sep 2006, 09:08
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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.

What is convex polygon
A convex polygon is a simple polygon whose interior is a convex set. The following properties of a simple polygon are all equivalent to convexity:

* Every internal angle is at most 180 degrees.
* Every line segment between two vertices of the polygon does not go
exterior to the polygon

To have minimun 8 sides a polygon should have sum of angles 180(n-2)
180(8-2) = 1080

(1). Exactly 3 of the interior angles of R are greater than 80 degrees.
Lets take the greatest value for these angles- 180 and smallest for other 5 -80.
180*3+80*5= 940 < 1080 ..SUFFI

(2) None of the interior angles of R are less than 60 degrees.
Angles can have any value ..NOT SUFFI

##### General Discussion
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14 Aug 2006, 10:57
1
kevincan 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.

The sum of the interion angles of the polygon = 180(n-2) where n is the number of sides in the polygon.

From (1) 180(n-2) >240
n-2 > 4/3
n>10/3
therefore n should be atleast 4 but we cannot say it is atleast 8

From(2) 180(n-2) >=60n
n-2 >= n/3
2n/3 >= 2
n >=3
we cannot say n is atleast 8

Combining both, we can still say it is atleast 4 but not 8.

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Re: R is a convex polygon. Does R have at least 8 sides? (1)  [#permalink]

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21 Apr 2014, 19:05
IMO is E.

A convex polygon with 4 sides can still have angles greater than 80 and less than 60. Experts advice on this?
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Re: R is a convex polygon. Does R have at least 8 sides?  [#permalink]

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08 Jun 2015, 00:50
1
I have another way to solve this question:

(1) Exactly 3 angles are greater than 80 degrees
Let's denote "x" is the remaining sides of the polygon, then the total sides is "x+3"
The sum of interior angles = (x+3-2)*180 = (x+1)*180 degress
(a) Each of 3 angles is < 180 degrees, so sum of x remaining angles is > (x+1)*180 - 3*180 = 180x - 360 degrees
(b) However, each of the x remaining angles is <80 -> total of x angles is <80x degress
From (a) and (b) => 180x - 360 <80x
=> x<3.6
=> total sides of polygon R < 6.6 =>(1) is SUFFICIENT

(2) None of the interior angles are less than 60 degrees
(c) Let's denote "y" is the total sides of polygon R => total of angles is (y-2)*180 degrees
From (2) and (c) => (y-2)*180 >= 60y
=>120y>=360
=> y>=3
=> (2) is NOT SUFFICIENT

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R is a convex polygon. Does R have at least 8 sides?  [#permalink]

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05 Nov 2015, 20:19
Q: Does R have 8 sides? ==> does sum of interior angles of R = (8-2)*180 = 1080?

St 1: Provides two facts: (Exactly 3 are greater than 80)

First fact: 3 angles are > 80. The range of of the sum of these 3 angles = 3*81 -- 3*180 = 243 ---540.

Second fact: The remaining angles are all <= 80. For 5 additional angles, the range of of the sum = 5*0 -- 5*80 = 0 --- 400.

For 8 angles, the range of of the sum of all angles = range#1 + range#2 = 243 ---940. 1080 falls outside the range and therefore, it is impossible for an 8-sided polygon to have 3 sides > 80 and 5 sides <= 80.

SUFFICIENT

Statement 2: Angles are all > 60.

Range of sums for 8 angles = 8*60 --- 8*180 = 480 --- 1440. 1080 falls in this range, and accordingly an 8-sided polygon is not excluded, but in the absence of additional information it is not confirmed. In other words it is possible for an 8-sided polygon to have all angles > 60, but polygons with angles > 60 are not necessarily 8 sided.

INSUFFICIENT
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Re: R is a convex polygon. Does R have at least 8 sides?  [#permalink]

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24 Jun 2017, 03:06
Wow what a question .
Took a lot of and still got it wrong .
A is the correct answer .
It uses the property of convex polygon that every interior angle in a convex polygon is less than 180 degrees.
And the trick part in this question is to assume that the polygon is 8 sided .
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Re: R is a convex polygon. Does R have at least 8 sides?  [#permalink]

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09 Mar 2019, 01:21
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Re: R is a convex polygon. Does R have at least 8 sides?   [#permalink] 09 Mar 2019, 01:21
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