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Math Help - circumference of a parallelogram

  1. #1
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    circumference of a parallelogram

    hey guys,

    i've been trying to calculate the circumference of a parallelogram by only knowing the length of the diagonals (which are 7 and 9 units).

    i drew many triangles, tried to use pythagoras and some trig stuff, but failed until now...
    thanks a lot for your help!
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  2. #2
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    Hello, ViresArcanum!

    Is there more information?
    As stated, there is no unique solution.


    i've been trying to calculate the perimeter of a parallelogram
    by only knowing the length of the diagonals (which are 7 and 9 units).
    Code:
                S                 R
                * - - - - - - - - *
               /  *           *  /
              /     * O   *     /
             /        *        / b
            /  4 *  θ  * 3  /
           /  *           *  /
          * - - - - - - - - *
          P        a        Q

    We have parallelogram PQRS with diagonals PR = 9,\:QS = 7
    . . intersecting at O\!:\;\;OP = 4.5,\:OQ = 3.5,\;\angle POQ = \theta

    Law of Cosines: . a^2 \;=\;4.5^2 + 3.5^2 - 2(4.5)(3.5)\cos\theta

    . . Hence: . a \;=\;\sqrt{32.5 - 31.5\cos\theta}

    It can be shown that: . b \;=\;\sqrt{32.5 + 31.5\cos\theta}


    Hence, the perimeter, 2a + 2b, is a function of \theta, not a constant.

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  3. #3
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    earboth's Avatar
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    Quote Originally Posted by ViresArcanum View Post
    hey guys,

    i've been trying to calculate the circumference of a parallelogram by only knowing the length of the diagonals (which are 7 and 9 units).

    i drew many triangles, tried to use pythagoras and some trig stuff, but failed until now...
    thanks a lot for your help!
    Unfortunately that isn't possible:

    The parallelogram consists of 2 congruent triangles or 2 pairs of congruent triangles. To determine one triangle unambiguously you need 3 pieces. Therefore you need the value of one more piece: an angle or the height or the length of one side.

    To demonstrate how the circumference can change without changing the initial conditions (length of diagonals are constant) I've made a sketch.
    Attached Thumbnails Attached Thumbnails circumference of a parallelogram-parallelogrammumfang.gif  
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  4. #4
    Super Member angel.white's Avatar
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    Those were both really great posts.
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