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Math Help - Challenging height problem with right pyramid based on kite plan with one right angle

  1. #1
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    Challenging height problem with right pyramid based on kite plan with one right angle

    Here is a problem that is driving me crazy. There ought to be a simple solution, but it eludes me.
    I've made a model from paper, and it is a feasible construction, that constrains h and point S:

    OX, OY, OZ are orthogonal XYZ axes.

    O is a fixed point. T lies on OX, R lies on OY, and P lies on OZ.

    S lies in the XY plane, so base kite ROTS is a planar quadrilateral (i.e. there is no hinge along OS).

    a, b, c, d & e are known (of fixed length).

    For convenience, TO > OR, and d > e.

    Also, c > b > a.

    What then is h (=OP) ???

    I can imagine raising point P along OZ, so points R, S and T on the base plane draw closer to O and lie in the XY plane thru O, while keeping a, b, c, d & e of fixed length(s), with R lying somewhere along OY, and T somewhere along OX.

    But I cannot see how to calculate h, (and thus solve the base kite dimensions and angles, as well as the rest of the pyramid).
    Yet all the points ought to be uniquely determined for the solution. My paper model works just fine, but I want a precise solution.

    Any clues to solve h, and thus the rest of the unknowns? I attach a pdf and a jpg of the construction (MyProblem.pdf and MyProblem.jpg).

    Your assistance would be greatly appreciated!

    Regards, Rob.
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  2. #2
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    Re: Challenging height problem with right pyramid based on kite plan with one right a

    As no one seems to be able to offer help on this, let me try to clarify the problem:

    Imagine a quadrilateral of paper PTSR is folded into two triangles along PS.

    The quadrilateral/ triangle pair is sat on a plane on edges RS and ST.

    The angle between the two planes of the triangles (the fold PS) is adjusted until PR and PT are in plan at right angles.

    How can this condition be solved to get the unknown information e.g. h, RO, OT, angles ROT and SOT etc.?

    I am only interested in solutions in the quadrant shown (with OT and OR positive) and with OP positive.

    Help, please! Thanks, cadastralrob.
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