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Math Help - Cabins B and G

  1. #1
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    Cabins B and G

    Cabins B and G are located on the shore of a circular lake, and cabin L is located near the lake. Point D is a dockon the lake shore and is collinear with cabins B and L. The road between cabins G and L is 8 miles long and is tangent to the lake. The path between cabin L and dock D is 4 miles long.
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  2. #2
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    Hello, magentarita!

    Your forgot to give us the question . . .


    Cabins B and G are located on the shore of a circular lake,
    and cabin L is located near the lake.
    Point D is a dock on the lake shore, collinear with cabins B and L.
    The road between cabins G and L is 8 miles long and is tangent to the lake.
    The path between cabin L and dock D is 4 miles long.
    Code:
             B    * * *
              o           *
            *   \           *
           *      \          *
                    \  x
          *           \       *
          *             \     *
          *               \   *
                            \  D
           *                 o
            *               *   \  4
              *           *       \
                  * o * - - - - - - o L
                    G         8

    The only questions we can answer are:
    . . What are the distances BD amd BL ?


    If that is indeed the question, we should know this theorem:

    . . If a tangent and a secant are drawn to a circle from an external point,
    . . the tangent is the mean proportional between the external segment
    . . of the secant and the entire secant.


    In this problem, we have: . \frac{LD}{LG} \:=\:\frac{LG}{LB} \quad\Rightarrow\quad \frac{4}{8} \:=\:\frac{8}{x+4}

    Then: . 4x + 16 \:=\:64 \quad\Rightarrow\quad 4x \:=\:48 \quad\Rightarrow\quad x \:=\:12


    Therefore: . BD = 12,\;\;BL = 16 miles.

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  3. #3
    MHF Contributor
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    ok

    Quote Originally Posted by Soroban View Post
    Hello, magentarita!

    Your forgot to give us the question . . .

    Code:
             B    * * *
              o           *
            *   \           *
           *      \          *
                    \  x
          *           \       *
          *             \     *
          *               \   *
                            \  D
           *                 o
            *               *   \  4
              *           *       \
                  * o * - - - - - - o L
                    G         8
    The only questions we can answer are:
    . . What are the distances BD amd BL ?


    If that is indeed the question, we should know this theorem:

    . . If a tangent and a secant are drawn to a circle from an external point,
    . . the tangent is the mean proportional between the external segment
    . . of the secant and the entire secant.


    In this problem, we have: . \frac{LD}{LG} \:=\:\frac{LG}{LB} \quad\Rightarrow\quad \frac{4}{8} \:=\:\frac{8}{x+4}

    Then: . 4x + 16 \:=\:64 \quad\Rightarrow\quad 4x \:=\:48 \quad\Rightarrow\quad x \:=\:12


    Therefore: . BD = 12,\;\;BL = 16 miles.
    Tell me, how do you make such beautiful geometric shapes with your keyboard?
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