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Thread: Inner diagonals?

  1. #1
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    Question Inner diagonals?

    How do you find inner diagonals of a cube? I was wondering that for a very long time.
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  2. #2
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    Re: Inner diagonals?



    Let a be the length of every side. D'B is an inner diagonal of the cube ABCDA'B'C'D'. B'D, A'C, C'A are also inner diagonals.

    DD' $\displaystyle \perp$ (ABCD), so DD' is perpendicular to DB (since DB is in (ABCD)), that means that BDD' is a right triangle:

    $\displaystyle D'D^2+DB^2=D'B^2 \Rightarrow D'B=\sqrt{D'D^2+DB^2}$

    ABD is a right triangle, so: $\displaystyle AD^2+AB^2=DB^2 \Rightarrow DB^2=2a^2$

    $\displaystyle D'B=\sqrt{a^2+2a^2}=\sqrt{3a^2}=a\sqrt3$
    Last edited by veileen; Feb 16th 2013 at 11:45 PM.
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