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Each side of a given polygon is parallel to either the X or
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Updated on: 14 Oct 2013, 00:34
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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, 05:12.
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
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14 Oct 2013, 00:13
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 +y2).
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
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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
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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
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29 Apr 2010, 04:10
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
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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..
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Angles.doc [24.5 KiB]
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Re: polygon
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29 Apr 2010, 05:44
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(n2). 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
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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 +y2).
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
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14 Mar 2016, 22:25
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
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05 May 2016, 11:53
My equations: 180(n  2) = 25(90) + 270(n  25). Ended up with: 90(46) = 90n
n = 46, n  25 = 21. Answer is 21.



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Re: Each side of a given polygon is parallel to either the X or
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26 Feb 2018, 06:49
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