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

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Each side of a given polygon is parallel to either the X or [#permalink]  16 Apr 2010, 05:12
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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

[Reveal] Spoiler:
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?).
[Reveal] Spoiler: OA

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Last edited by Bunuel on 14 Oct 2013, 00:34, 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]  19 Apr 2010, 14:04
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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Re: Polygon with concave and convex corners [#permalink]  19 Apr 2010, 14:16
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]  29 Apr 2010, 04:10
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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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Re: polygon [#permalink]  29 Apr 2010, 05:17
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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Re: polygon [#permalink]  29 Apr 2010, 05:44
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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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Re: Polygon with concave and convex corners [#permalink]  14 Oct 2013, 00:13
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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]  14 Feb 2014, 04:58
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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Re: Each side of a given polygon is parallel to either the X or [#permalink]  18 May 2015, 21:59
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Re: Each side of a given polygon is parallel to either the X or   [#permalink] 18 May 2015, 21:59
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