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Director  Joined: 25 Aug 2007
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Each side of a given polygon is parallel to either the X or  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 56% (02:29) correct 44% (02:05) wrong based on 144 sessions

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Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the number of convex corners in such a polygon is 25, the number of concave corners must be:

(A) 20
(B) 0
(C) 21
(D) 22
(E) 23

3. 21
In this kind of polygon, the number of convex angles will always be exactly 4 more
than the number of concave angles (why?).

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Originally posted by ykaiim on 16 Apr 2010, 06:12.
Last edited by Bunuel on 14 Oct 2013, 01:34, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Polygon with concave and convex corners  [#permalink]

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1
Let the number of convex corner be x and the number of concave corners be y.
Therfore the sum of the angles of the polygon would be 90*x +270*y.

Also the sum of the angles can also be given by 180*(x +y-2).

Therefore,
90*x + 270*y = 180*(x + y - 2)
or, 90*x - 90*y = 360
or x - y = 4.

Therefore if x = 25, y = 21.
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Re: Polygon with concave and convex corners  [#permalink]

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I had to visualize this one. In a polygon with sides parallel to the x or y axis, the smallest number of sides would be four, making a rectangle (4 concave corners, 0 convex corners). As you "add" corners, you can do it one of two ways: either by taking a "bite" out of a corner taking away one concave corner, and replacing it with one convex corner and two concave corners, or taking a bite out of a side adding two concave and two convex corners. Either way, you are still add a 1:1 ratio of corners with each addition, plus the four corners you had at the beginning.So you will always have four more concave corners than convex corners.

If there is a mathmatical way to figure this out I would love to hear it, but visuallizing it in this way is the fastest way I can think of.
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GMAT 1: 670 Q49 V31 GMAT 2: 710 Q50 V35 Re: Polygon with concave and convex corners  [#permalink]

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mmphf wrote:
I had to visualize this one. In a polygon with sides parallel to the x or y axis, the smallest number of sides would be four, making a rectangle (4 concave corners, 0 convex corners). As you "add" corners, you can do it one of two ways: either by taking a "bite" out of a corner taking away one concave corner, and replacing it with one convex corner and two concave corners, or taking a bite out of a side adding two concave and two convex corners. Either way, you are still add a 1:1 ratio of corners with each addition, plus the four corners you had at the beginning.So you will always have four more concave corners than convex corners.

If there is a mathmatical way to figure this out I would love to hear it, but visuallizing it in this way is the fastest way I can think of.

visualizing is the best way. pls check the bolded text above, it should be vice versa
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Each side of a given polygon is parallel to either the X or  [#permalink]

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Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90o or concave if the internal angle is 270o. If the number of convex corners in such a polygon is 25, the number of concave corners must be:

(1)20
(2)0
(3)21
(4)22
(5)23
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I also think the answer is 21...

In the figure attached

1,2,3,4 so on represent convex angles...

1`,2`,3`... so on represent concave angles... which will be 21..
Attachments Angles.doc [24.5 KiB]

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lanka1 wrote:
Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90o or concave if the internal angle is 270o. If the number of convex corners in such a polygon is 25, the number of concave corners must be:

(1)20
(2)0
(3)21
(4)22
(5)23

Sum of all the angles is 180(n-2).
Let the total no of 90 deg angle be x
Let the total no of 270 deg angle be y

n= x+y, so sum of all the angles is 180(x+y -2)

so, 90x + 270y = 180(x+y -2)
90x + 270y = 180x+ 180y - 360
90y = 90x -360

Given x=25
y= (90*25 - 360)/90
y= 21
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Posts: 142
Re: Polygon with concave and convex corners  [#permalink]

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Yes, this seems to be the best way to solve the problem. I tried to visualize the polygon and ended up with correct answer as 21.

End deduction was that the number of convex corners of such polygon are always 4 more than number of concave corners of the same polygon.

nitesh181989 wrote:
Let the number of convex corner be x and the number of concave corners be y.
Therfore the sum of the angles of the polygon would be 90*x +270*y.

Also the sum of the angles can also be given by 180*(x +y-2).

Therefore,
90*x + 270*y = 180*(x + y - 2)
or, 90*x - 90*y = 360
or x - y = 4.

Therefore if x = 25, y = 21.

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Each side of a given polygon is parallel to either the X or  [#permalink]

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ykaiim wrote:
Each side of a given polygon is parallel to either the X or the Y axis. A corner of such a polygon is said to be convex if the internal angle is 90° or concave if the internal angle is 270°. If the number of convex corners in such a polygon is 25, the number of concave corners must be:

(A) 20
(B) 0
(C) 21
(D) 22
(E) 23

3. 21
In this kind of polygon, the number of convex angles will always be exactly 4 more
than the number of concave angles (why?).

It is because of the geometric pattern.
for a square you have 4 convex corners.
If you want to increase the number of sides you have to make an opening on one side. When you open one hole and draw lines parallel to x and y axis, it adds 2 convex and 2 concave corners (check it!). This pattern goes on on and makes a difference of 4, like, (4,0), (6,2), (8,4) and so on.
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Re: Each side of a given polygon is parallel to either the X or  [#permalink]

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1
My equations: 180(n - 2) = 25(90) + 270(n - 25).
Ended up with: 90(46) = 90n

n = 46, n - 25 = 21.
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Re: Each side of a given polygon is parallel to either the X or  [#permalink]

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_________________ Re: Each side of a given polygon is parallel to either the X or   [#permalink] 03 May 2019, 06:26
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