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Math Help - inscribe square into semicircle?

  1. #1
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    inscribe square into semicircle?

    How do you inscribe a square into a given semi-circle?
    and i need the proof also.
    please and thanks.
    Last edited by Pabo258; February 16th 2009 at 12:30 AM.
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  2. #2
    Member Nacho's Avatar
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    Inscribe a rectangle of wide X and long 2X in a circunference...
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  3. #3
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    Hello, Pabo258!

    How do you inscribe a square into a given semicircle?
    Are you talking about a construction using Eucliden tools?

    Here is one method . . .
    Code:
                            C * -
                             /| :
                            / | :
                           /  | :
              R   * * *  P/   | 2r
               * - - - - o    | :
            *  :        /:  * | :
           *   :       / :   *| :
               :      /  :    | :
          *    :     /   :    * :
          *----+----*----*----* -
          B         O    Q    A
          : -  r  - : -  r  - :

    We have a semicircle with center O and radius OA = OB = r.

    At A erect a perpendicular to diameter AB
    . . and measure off a distance 2r\!:\;AC = 2r

    Draw line segment OC, intersecting the semicircle at P.

    From P drop a perpendicular PQ to diameter AB.

    From P draw a horizotal line, intersecting the semicircle at R.

    Segments PQ and PR determine the inscribed square.


    I'll let someone else provide the proof . . .

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  4. #4
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    Proof

    Does anyone know how to prove this?
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  5. #5
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    On Soroban's clever diagram

    |AC|=2r

    |OA|=r

    |AC|=2|OA|

    Triangles OPQ and OCA are similar.

    Triangles OCA is a magnified version of triangle OPQ.

    Therefore

    |QP|=2|OQ|

    Hence |QP|=|PR| and the shape is a square.
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  6. #6
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    The idea is to draw a line from the semicircle centre
    with a slope of 2 and mark the point where the line touches the semicircle circumference.
    Calculating the inverse tangent of 2 gives you the acute angle the line makes
    with the x-axis.
    However, simplest is to pick a horizontal value x,
    and a vertical value 2x directly above x.
    Locate that point and draw a line from the semicircle centre through it.
    Picking x=r and 2x=2r is far neatest.
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  7. #7
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    Area of inscribed square in semicircle radius r = x
    Area of inscribed square in circle radius r = y
    x/y = 2/5
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