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Math Help - Geometry Word Problem (Arcs)

  1. #1
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    Geometry Word Problem (Arcs)

    A bridge over a river has a shape of a circular arc. The span of the bridge is 24 meters. (The span is the length of the chord of the arc.) The midpoint of the arc is 4 meters higher than the endpoints. What is the radius of the circle that contains this arc?

    The radius = 20, I'm not sure how to get to that answer, though.


    Thanks in advance.
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  2. #2
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    Lexington, MA (USA)
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    Hello, dvh9!

    A bridge over a river has a shape of a circular arc.
    The span of the bridge is 24 meters. (The span is the length of the chord of the arc.)
    The midpoint of the arc is 4 meters higher than the endpoints.
    What is the radius of the circle that contains this arc?
    Code:
                    C
                  * * *
              *    4|     *
          A *-------o-------* B
           *  * 12  |D    *  *
                *   |   * R
          *       * | *       *
          * - - - - * - - - - *
                    O

    The center of the circle is O.
    The radius is: . R \,=\,OA \,=\,OB \,=\,OC.
    CD = 4, hence: . OD = R - 4
    AD \,=\,DB \,=\,12


    In right triangle ODB\!:\;\;OD^2 + DB^2 \:=\:OB^2

    So we have: . (R-4)^2 + 12^2 \:=\:R^2 \quad\Rightarrow\quad R^2 - 8R + 16 + 144 \:=\:R^2

    . . -8R \:=\:-160 \quad\Rightarrow\quad R \:=\:20

    Therefore, the radius of the arc is 20 meters.

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  3. #3
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    Wonderful

    Wonderful description, thanks!
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