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Homework Help: Prove that the ratio of the length of a diagonal to that of its corresponding side

  1. Feb 6, 2012 #1
    Pentagon diagonal - parallel ratio proof

    1. The problem statement, all variables and given/known data


    Suppose P is a convex pentagon such that each diagonal is parallel to one side. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio

    2. Relevant equations



    3. The attempt at a solution
    I think they mean a pentagon where each vertices is connected by an edge. So there is a pentagon in the middle surrounded by obtuse and acute triangles. So if you label the sides of the original pentagon each side is say a+b, with the bases of the acute triangles are say b and the sides are say a. Then the ratio of the paralleled sides is a+b:2a+b? Not really sure how i am meant to prove this though?
     
    Last edited: Feb 6, 2012
  2. jcsd
  3. Feb 6, 2012 #2

    Mark44

    Staff: Mentor

    Re: Pentagon diagonal - parallel ratio proof

    No, I don't think so. Each pair of adjacent vertices is connected by an edge, but nonadjacent vertices are not connected.
    No. It's just a pentagon, a five-sided figure. It's a convex pentagon, which means that the interior angles are all less than 180°.
    The pentagon is not surrounded by triangles of any kind. I don't know where you got that idea.
    KEEP IT SIMPLE. Just draw a pentagon, and draw some of the diagonals, keeping in mind what it says about them being parallel to another side of the pentagon.
     
  4. Feb 7, 2012 #3
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    Oh I meant the triangles are in the interior, when you draw the diagonals. So if a pentagons vertices are labelled 1 to 5, then say the length of 3-4 is a+b, then the parallel diagonal is length 2a+b?
     
  5. Feb 7, 2012 #4
    Re: Pentagon diagonal - parallel ratio proof

    .d9n.,

    would it be the case that this convex pentagon would have to
    be equilateral?


    Furthermore, would it be the case that this convex pentagon
    would have to be a regular pentagon?
     
  6. Feb 7, 2012 #5
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    im not sure this is all the info i have
     
  7. Feb 7, 2012 #6

    Mark44

    Staff: Mentor

    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    Why are you setting the length of the edge between vertice 3 and 4 to a + b? The parallel diagonal would be between vertices 2 and 5. Why would you think this would be 2a + b?

    What you are doing is very confusing.
     
  8. Feb 7, 2012 #7
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    yer i know, this is where i got the triangles and ratios from

    http://www.jimloy.com/geometry/pentagon.htm [Broken]
     
    Last edited by a moderator: May 5, 2017
  9. Feb 7, 2012 #8

    Mark44

    Staff: Mentor

    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    The figure in the page in the link appears to be an equilateral pentagon, which is information you are not given. From the information you are given, you cannot assume that all the sides are equal. It may be that they are, but you would need to show that, using geometry and trig.
     
  10. Feb 7, 2012 #9
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    oh right, any ideas then on how i do that, im a bit lost
     
  11. Feb 7, 2012 #10
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    Not only does the pentagon in the link appear to be equilateral, it also
    appears to be equiangular.
     
  12. Feb 8, 2012 #11
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    so any ideas on how to go about proving this? i have no idea
    thanks
     
  13. Feb 8, 2012 #12

    Mark44

    Staff: Mentor

    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    I would start by drawing a pentagon, and a couple of its diagonals. Use the given information that each diagonal is parallel to one side. This means that a given side of the pentagon, its two adjacent sides, and the diagonal, form a trapezoid. See where that takes you.
     
  14. Feb 9, 2012 #13
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    ok so if we label each trapezoid:

    if we assign lengths to each edge

    ab=v, bc=w, cd=x, de=y, ea=z,

    now the diagonals

    ac=f, ad=g, be=h, bd=i, ce=j

    1. abcd ratio = bc/ad = w/g
    2. bcde ratio = cd/be = x/h
    3. abde ratio = ae/bd = z/i
    4. acde ratio = de/ac = y/f
    5. abce ratio = ab/ce = v/j

    not sure how to prove all the ratios are the same without it being equilateral
     
  15. Feb 9, 2012 #14

    Mark44

    Staff: Mentor

    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    You need to use the given information that each diagonal is parallel to one of the sides. From that, you need to establish relationships between the angles formed by the interior triangles.
     
  16. Feb 9, 2012 #15
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    do u mean by putting diagonals in the trapezoid?
    unsure how to do this if i cant assume the the shape is equilateral, as surely the angles could be anything depending on the shape of the original pentagon.
     
  17. Feb 9, 2012 #16
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    if your make the middle of the pentagon 'o' say then, its 360n degrees around it divide by 5 leaves each angle aob, boc, cod, doe, eoa is 72 degrees?
    making the angles abc bcd cde dea eab all 108? dont know if this is of any help though?
     
  18. Feb 9, 2012 #17
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    which potentially means angle ead and similar are 36 degrees?
     
  19. Feb 9, 2012 #18

    Mark44

    Staff: Mentor

    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    Only if it's a regular pentagon, which is information you're not given.
     
  20. Feb 9, 2012 #19
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    that's what i thought, so im unsure of how to work out the angles if we dont know what type of pentagon it is
     
  21. Feb 9, 2012 #20
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    surely the angles will change depending on what type of pentagon it is, i.e. isosceles, equilateral, or neither. I've looked all over trying to find a way to do it, yet no luck
     
  22. Feb 9, 2012 #21

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    You can't assume that the pentagon is equilateral. It doesn't have to be equilateral for this theorem to be true.

    You can't pick some arbitrary convex pentagon either. It's likely that such a pentagon would not fulfill the condition that each diagonal is parallel to one side.

    This looks to me to be very difficult to prove -- unless there is some nice trick.
     
    Last edited: Feb 9, 2012
  23. Feb 9, 2012 #22
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    can i ask what theorem you are referring to?

    Any hints on where to look for this 'nice trick', is it enclosed in the theorem you refer to?
     
  24. Feb 9, 2012 #23
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    so when a line intersects the two parallel lines, where the top line is intersected the the interior angle is say 'a' so the interior angle on the bottom line is 180-a?
     
  25. Feb 9, 2012 #24
    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    but then i still dont know how that helps me as i cant assume that another intersecting line intersects the parallel line at the same angle. im stuck
     
  26. Feb 9, 2012 #25

    SammyS

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    Re: Prove that the ratio of the length of a diagonal to that of its corresponding sid

    I was referring to the conjecture you're trying to prove here.

    I have no idea about where to look for such a trick.
     
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