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Math Help - [SOLVED] Geometry Treasure Pirate Problem

  1. #1
    johnkid
    Guest

    [SOLVED] Geometry Treasure Pirate Problem

    My teacher assigned this very confusing, at least to me, problem.

    (Summed up by me.): Some people found a treasure map, authenticated it and now looking for it. Here's what it says:

    "Start from the tree, walk directly to the falcon-shaped rock. Count your paces as you walk. Turn a quarter-circle to the right and walk the same number of paces. When you reach the end put a stick in the ground. Return to the tree and likewise walk to the owl-shaped rock, again counting your paces. Turn a quarter-circle to the left, walk the same number of paces, and put another stick in the ground. Connec the two sticks with a rope. The treasure is beneath the midpoint of the rope."

    HOWEVER, the tree was no longer on the island. All that was left was the two weird shaped rocks. The people gave up looking for the treasure.

    Did the people give up too soon? Is it possible to find the treasure without the tree? Explain and give a rough sketch.




    My answer is no. I think there are way to many possiblities. I drew out a lot of sketches and my head starts to hurt. So, if you know of the answers would you please explain it to me? Thanks
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
    Posts
    1,631
    As it is summed up by you, or as posted, I am seeing millions of possible answers.

    Can you write your teacher's question exactly as she/he spoke it to you? Maybe we can get a different summary, and a unique solution too.

    This portion:
    "Start from the tree, walk directly to the falcon-shaped rock. Count your paces as you walk. Turn a quarter-circle to the right and walk the same number of paces. When you reach the end put a stick in the ground. Return to the tree and likewise walk to the owl-shaped rock, again counting your paces. Turn a quarter-circle to the left, walk the same number of paces, and put another stick in the ground. Connec the two sticks with a rope. The treasure is beneath the midpoint of the rope."
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  3. #3
    Super Member

    Joined
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    Lexington, MA (USA)
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    Hello, John!


    Some people found a treasure map, authenticated it and now looking for it.
    Here's what it says:

    "Start from the tree, walk directly to the falcon-shaped rock.
    Count your paces as you walk.
    Turn a quarter-circle to the right and walk the same number of paces.
    When you reach the end put a stick in the ground.

    Return to the tree and likewise walk to the owl-shaped rock, again counting your paces.
    Turn a quarter-circle to the left, walk the same number of paces,
    and put another stick in the ground. Connect the two sticks with a rope.
    The treasure is beneath the midpoint of the rope."

    HOWEVER, the tree was no longer on the island.
    All that was left was the two weird-shaped rocks.
    The people gave up looking for the treasure.

    Did the people give up too soon?
    Is it possible to find the treasure without the tree? . . . . Yes!

    This is a classic problem.
    Code:
                             Q
                        X    o
                 P      o    |*
                 o      |    | *
                *|      |    |  *
               * |      |    |   *
              *  |      |    |    *
             *   |      |    |     *
            *    |      |    |      *
           *     |    B |    |       *
        F o------*----*-*----*--------o O
             *   A    | C    D    *
                *     |       *
                   *  |   *
                      o
                      T

    We start at T, walk to F, turn 90 to the right,
    . . and walk to point P, where FP = TF.

    We start at T, walk to O, turn 90 to the left,
    . . and walk to point Q, where OQ = TO.

    The treasure is buried at X, the midpoint of PQ.

    Draw perpendiculars PA,\:TB,\:XC,\:QD to FO.


    Since X is the midpoint of PQ, XC is the average of PA and QD.
    . . That is: . XC \:=\:\frac{PA + QD}{2} .[1]

    We find that: . \Delta PAF \cong \Delta FBT\quad\Rightarrow\quad PA = FB .[2]

    We find that: . \Delta QDO \cong \Delta OBT\quad\Rightarrow\quad QD = BO .[3]

    Substitute [2] and [3] into [1]: . XC\:=\:\frac{FB + BO}{2} \:=\:\frac{1}{2}FO

    Hence, the "height" of X is half of the distance between the two rocks.


    Since X is the midpoint of PQ, it follows that: C is the midpoint of AD.

    From the congruent right triangles: . FA = TB and DO = TB
    . . That is, the two end segments of FO are equal.
    Hence, C is the midpoint of FO.


    To locate the treasure X, we don't need the tree.

    Locate C, the midpoint of FO.
    From C, walk perpendicular to FO a distance of \frac{1}{2}FO
    . . and there is the treasure!

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