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Math Help - Proved that

  1. #1
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    Proved that

    Last edited by dapore; April 20th 2010 at 09:56 AM.
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  2. #2
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    Hello, dapore!

    In any triangle ABC with angles A,B,C and sides a,b,c\!:

    . . . b\cos A + a\cos B \:=\:c
    Code:
                  C
                  o
                 *| *
                * |   *
               *  |     *
            b *   |       * a
             *    |         *
            *     |           *
           *      |             *
        A o   *   o   *   *   *   o B
              x   D       y
          : - - - - - c - - - - - :

    Construct altitude CD to side AB.
    Let x = AD,\;Y = DB

    In right triangle CDA\!:\;\;\cos A \,=\,\frac{x}{b} \quad\Rightarrow\quad x \,=\,b\cos A .[1]

    In right triangle CDB\!:\;\;\cos B \,=\,\frac{y}{a} \quad\Rightarrow\quad y \,=\,a\cos B .[2]


    Note that: . x + y \:=\:c .[3]


    Substitute [1] and [2] into [3]: . b\cos A + a\cos B \;=\;c
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  3. #3
    Member
    Joined
    Dec 2009
    Posts
    79
    Quote Originally Posted by Soroban View Post
    Hello, dapore!

    Code:
                  C
                  o
                 *| *
                * |   *
               *  |     *
            b *   |       * a
             *    |         *
            *     |           *
           *      |             *
        A o   *   o   *   *   *   o B
              x   D       y
          : - - - - - c - - - - - :
    Construct altitude CD to side AB.
    Let x = AD,\;Y = DB

    In right triangle CDA\!:\;\;\cos A \,=\,\frac{x}{b} \quad\Rightarrow\quad x \,=\,b\cos A .[1]

    In right triangle CDB\!:\;\;\cos B \,=\,\frac{y}{a} \quad\Rightarrow\quad y \,=\,a\cos B .[2]


    Note that: . x + y \:=\:c .[3]


    Substitute [1] and [2] into [3]: . b\cos A + a\cos B \;=\;c
    Thanks Soroban
    just for you


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