Results 1 to 5 of 5

Math Help - Convex 10-gon

  1. #1
    Newbie
    Joined
    Jan 2009
    From
    Charleston, IL
    Posts
    18

    Convex 10-gon

    I have a couple of proofs about a convex 10-gon that I need help with.

    "1. Prove there exists a convex 10-gon that has 3 acute, interior angles.

    2. Prove there does not exist a convex 10-gon that has 4 acute, interior angles."
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,614
    Thanks
    1581
    Awards
    1
    Do you understand that the sum of the interior angles is 8\pi?
    If there were three acute angles the remaining seven angles would sum to more than \frac{13\pi}{2}.
    Is that possible? Recall that each of those angles would between 0\;\&\;\pi

    BUT, what would happen if there were four acute angles?
    What is the possible sum of the remaining six angles?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jan 2009
    From
    Charleston, IL
    Posts
    18
    Quote Originally Posted by Plato View Post
    Do you understand that the sum of the interior angles is 8\pi?
    If there were three acute angles the remaining seven angles would sum to more than \frac{13\pi}{2}.
    Is that possible? Recall that each of those angles would between 0\;\&\;\pi

    BUT, what would happen if there were four acute angles?
    What is the possible sum of the remaining six angles?
    Well I do now. I've been sick all weekend but here's what work I got from your post.

    For the first one it is possible because \frac{13\pi}{2} is 20.42 and taking 7pi can give you 21.99. (My LaTeX sucks)

    I don't know the properties of 10-gons but taking 3/10 of 8 would be 2.4 so where did you get 13/2 instead of 5.6? (I took the pi out to make it simplier but 5.6pi would be what I thought would be the sum you need to go over.

    This is the question that has me stuck on how to get what the sum shouldn't be on the second question. (trying to disprove the sums)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,614
    Thanks
    1581
    Awards
    1
    Quote Originally Posted by maddmaxx46 View Post
    I don't know the properties of 10-gons but taking 3/10 of 8 would be 2.4 so where did you get 13/2 instead of 5.6?
    If you don’t know the properties of 10-gons, I think it is almost impossible to attempt this problem.
    Have a look at this webpage: Polygon -- from Wolfram MathWorld
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2009
    From
    Charleston, IL
    Posts
    18
    Quote Originally Posted by Plato View Post
    If you donít know the properties of 10-gons, I think it is almost impossible to attempt this problem.
    Have a look at this webpage: Polygon -- from Wolfram MathWorld
    Actually looking at this problem I think it's easier than I expected. 1440^\circ is the total sum of the interior angles. If 3 angles are acute, then the sum of the remaining angles should be more than 1170^\circ. So by taken 1170 divided by 7 I got 167.14 which is below 180^\circ keeping it convex.

    For the second proof the remaining angles should be more than 1080^\circ and by taking that divided by 6 I get 180 and going past 180 would make the decagon concave.

    I hope this is what you were hinting at in your first post.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Convex Set
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 23rd 2010, 07:00 PM
  2. The union of two convex sets is not convex
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: January 30th 2010, 03:23 PM
  3. Proving that max{0,f(x)} is convex if f is convex
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 5th 2009, 06:16 AM
  4. Proving that f^2 is convex if f is convex and f>=0
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 3rd 2009, 09:51 AM
  5. Convex help please!
    Posted in the Geometry Forum
    Replies: 3
    Last Post: September 23rd 2008, 01:40 PM

Search Tags


/mathhelpforum @mathhelpforum