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Math Help - Parallelogram Question

  1. #1
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    Parallelogram Question

    I feel really dumb asking this..but whatever

    one length of the pgram is 7 the other is 9..the long diagonal is 14..whats the short diagonal
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  2. #2
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    Hello, stones44!

    We will need Trignometry for this one . . . the Law of Cosines.


    One side of the parallelogram is 7, the other is 9.
    The long diagonal is 14.
    What is the short diagonal?
    Code:
               A         9         B
                * - - - - - - - - *
               /              *  /
              /           *     /
            7/        *        / 7
            /     *    14     /
           /  *              /
          * - - - - - - - - *
         D         9         C
    We have parallelogram ABCD with: AB = DC = 9,\;AD = BC = 7,\;BD = 14
    . . and we want the length of diagonal AC.


    First, find \angle A.

    . . In \Delta ABD\!:\;\;\cos A \;=\;\frac{7^2+9^2-14^2}{2(7)(9)}\;=\;-\frac{11}{21} \quad\Rightarrow\quad A \:=\:121.5881355^o

    Then \angle D \:=\:\angle ADC \:=\:180^o - 121.5881355^o \:=\:58.41186449^o


    In \Delta ADC\!:\;\;AC^2\;=\;7^2+9^2-2(7)(9)\cos58.41186449^o \;=\;63.99999999

    Therefore: . AC \:=\:\sqrt{63.99999999} \:\approx\:8

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  3. #3
    MHF Contributor

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    Suppose that vectors a,\,b are two adjacent sides of the parallelogram.
    Then a + \,b\quad \& \quad a - b would represent the two diagonals.
    We know that \left\| {a + b} \right\|^2  = \left\| a \right\|^2  + 2a \cdot b + \left\| b \right\|^2 \quad \& \quad \left\| {a - b} \right\|^2  = \left\| a \right\|^2  - 2a \cdot b + \left\| b \right\|^2
    So \left\| {a + b} \right\|^2  + \left\| {a - b} \right\|^2  = 2\left\| a \right\|^2  + 2\left\| b \right\|^2.

    Use that to find the length of the second diagonal.
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