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Math Help - **Hexagon Help**

  1. #1
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    **Hexagon Help**

    I am stuck with this question, can anyone help?

    Hexagon ABCDEF has the following properties:
    i. diagonals AC, CE and EA are all the same length
    ii. angles ABC and CDE are both 90 degrees
    iii. all the sides of the hexagon have lengths which are different integers

    a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
    b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

    Any help would be much appreciated

    thanks
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  2. #2
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    Quote Originally Posted by Chuck_3000
    I am stuck with this question, can anyone help?

    Hexagon ABCDEF has the following properties:
    i. diagonals AC, CE and EA are all the same length
    ii. angles ABC and CDE are both 90 degrees
    iii. all the sides of the hexagon have lengths which are different integers

    a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
    b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

    Any help would be much appreciated

    thanks
    First the diagram: see attachment.

    Now Part (a):

    Now \triangle\ ABC and \triangle\ CDE are right triangles with hypotenuses of length
    \sqrt{85}, so for \triangle ABC:

    <br />
AB^2+BC^2=85<br />

    with AB and BC integers, and trail and error shows that the pair
    of lengths (AB,BC) is one of (2,9),\ (6,7),\ (9,2),\ (7,6).

    The same argument applies to \triangle CDE.

    So for part (a) we have the minimum of AB+BC+CD+DE=24
    (each pairs (AB,BC),\ (CD,DE) is one of (2,9),\ (6,7),\ (9,2),\ (7,6)
    but with none of the sides equal, so these sides is one of 2,6,7,9 in some order, without repetition)

    Also EF+FA cannot be 10 or less (trial and error shows this can't be
    10, and it must be greater than 9 as it must be greater than \sqrt{85}).
    But EF+FA can be 11.

    So the minimum perimeter is 24+11=35.

    RonL
    Attached Thumbnails Attached Thumbnails **Hexagon Help**-hex.jpg  
    Last edited by CaptainBlack; May 1st 2006 at 01:12 AM.
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  3. #3
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    Quote Originally Posted by Chuck_3000
    I am stuck with this question, can anyone help?

    Hexagon ABCDEF has the following properties:
    i. diagonals AC, CE and EA are all the same length
    ii. angles ABC and CDE are both 90 degrees
    iii. all the sides of the hexagon have lengths which are different integers

    a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
    b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

    Any help would be much appreciated

    thanks
    Hello,

    to a)
    You have to do with right triangles. The squares of the legs must be 85, that means, you have to split 85 into 2 squares. Thus:
    AB = 2
    BC = 9

    DC = 6
    ED = 7

    Now you are looking for integers, which are not present in the list of legs above(1, 3, 4, 5, 8). The sum of these integers must exceed √(85) approximately 9.22. Therefore you have as possible combination:

    EF = 3, AF = 8
    EF = 4, AF = 8
    EF = 5, AF = 8

    I'm awfully sorry, but I can't offer you any help for part b).

    Greetings

    EB
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    thank you so much guys!

    CaptainBlack, how do you get those images on there?
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  5. #5
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    Quote Originally Posted by Chuck_3000
    I am stuck with this question, can anyone help?

    Hexagon ABCDEF has the following properties:
    i. diagonals AC, CE and EA are all the same length
    ii. angles ABC and CDE are both 90 degrees
    iii. all the sides of the hexagon have lengths which are different integers

    a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
    b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?

    Any help would be much appreciated

    thanks
    part (b)

    Just an idea:

    The smallest length for AC is the square root of the smallest number
    that can be written as the sum of two squares in two different ways
    with no common square among the four.

    I will think about this later after we have finished playing HeroScape here

    RonL
    Last edited by CaptainBlack; May 1st 2006 at 03:37 AM.
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  6. #6
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    Quote Originally Posted by Chuck_3000
    thank you so much guys!

    CaptainBlack, how do you get those images on there?
    I use PowerPoint as a drawing package, then capture the image as
    a bitmap into Paint Shop Pro (or any other photo editor) then
    resize and save as a JPEG.

    Then when composing a post I upload the image from the manage
    attachments dialog.

    RonL
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  7. #7
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    Quote Originally Posted by Chuck_3000
    I am stuck with this question, can anyone help?

    Hexagon ABCDEF has the following properties:
    i. diagonals AC, CE and EA are all the same length
    ii. angles ABC and CDE are both 90 degrees
    iii. all the sides of the hexagon have lengths which are different integers

    a) What is the minimum perimeter of ABCDEF if AC = √85 (sqr root of 85)
    b) What is the smallest length of AC for which ABCDEF has all these properties? What is the minimum perimeter in this case?...
    Hello,

    to b) How about
    \sqrt{65}=1^2 + 8^2 = 4^2 + 7^2

    Greetings

    EB
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  8. #8
    Forum Admin topsquark's Avatar
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    An almost entirely unrelated question.

    Doesn't "hexagon" imply equal sides? So wouldn't the question be asking about a "hexalateral" or some such?

    Just curious.

    -Dan
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  9. #9
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    Quote Originally Posted by topsquark
    An almost entirely unrelated question.

    Doesn't "hexagon" imply equal sides? So wouldn't the question be asking about a "hexalateral" or some such?

    Just curious.

    -Dan
    No hexagon=six sided planar figure, with all sides equal we are talking about
    a regular hexagon.

    RonL
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  10. #10
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack
    No hexagon=six sided planar figure, with all sides equal we are talking about
    a regular hexagon.

    RonL
    Now I just feel silly. I knew that. (Ahem!)

    -Dan
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  11. #11
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    Quote Originally Posted by CaptainBlack
    No hexagon=six sided planar figure, with all sides equal we are talking about
    a regular hexagon.

    RonL

    I can make that a large hexagon for an extra 50 cents,
    and would you like fries with that?

    RonL
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  12. #12
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CaptainBlack
    I can make that a large hexagon for an extra 50 cents,
    and would you like fries with that?

    RonL
    SuperSize me!

    Wait, don't. I'm big enough already!

    -Dan
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