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Math Help - Show constructibility

  1. #1
    Member courteous's Avatar
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    Question Show constructibility

    "Constructible lengths are those that can be built from known lengths by the operations of +, -, \times, \% and \sqrt."

    Show, using similar triangles, that if lengths  l_1 and  l_2 are constructible, then so are l_1l_2 and l_1/l_2.
    How do you go about it?




    How do you get a proper minus sign in LaTeX? I've tried "-", "$-$", "--", ... nothing works.
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  2. #2
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    Construct a triangle PQR with PQ of length l_{1},  QR of length 1 and RQ of any length at all (so long as PQR is a triangle).
    Now construct a line Q'R', parallel to QR and of length l_{2}, followed by lines P'Q' and P'R' parallel to there none dash equivalents.
    The two triangles will be similar and if you work out the length of P'Q', it will be seen to be l_{1}l_{2}.
    A similar routine can be used to construct a line of length l_{1}/l_{2}.
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