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Math Help - Similarity Problem

  1. #1
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    Similarity Problem

    Hi, in this problem, I'm not sure what the diagram looks like for a start, also I'm not sure how to solve it. I think congruency/similarity is involved.

    Let XYZ be a triangle with a right angle at Z. The midpoint of hypotenuse XY is twice as far from side YZ as it is from side XZ. Find the length of side XZ, if XY = 10mm
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  2. #2
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    Hello, BG5965!

    If you sketch the diagram, it's quite simple . . .


    Let XYZ be a triangle with a right angle at Z.
    The midpoint M of hypotenuse XY is twice as far from side YZ as it is from side XZ.
    Find the length of side XZ, if XY = 10 mm
    Let d = distance from M to XZ.
    Code:
        X *
          |\
          | \
       2d |  \ 5
          |   \
          |  d \  M
          * - - *
          |\    :\
          | \   : \
       2d |  \  :  \ 5
          |   \ :2d \
          |    \:    \
        Z *-----*-----* Y
             d     d
    The diagram is filled with congruent right triangles!

    Using Pythagorus on \Delta XYZ\!:\;\;(2d)^2 + (4d)^2 \:=\:10^2 \quad\Rightarrow\quad 4d^2 + 16d^2 \:=\:100

    . . 20d^2 \:=\:100 \quad\Rightarrow\quad d^2 \:=\:5 \quad\Rightarrow\quad d \:=\:\sqrt{5}


    Therefore: . XZ \:=\:4d \;=\;4\sqrt{5}

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  3. #3
    Member
    Joined
    May 2008
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    But with XY = 10, that diagram isn't correct, because although you put d and 2d, when I draw that diagram out with a ruler, it doesn't work!!!

    Is there a special way of doing it or do I assume 'not to scale'?

    Thanks
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