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Math Help - Divide a triangle into 2 parts equal area by a line parallel to its base.

  1. #1
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    Divide a triangle into 2 parts equal area by a line parallel to its base.

    A proof or brief explanation why would be nice too! Thank you!!
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  2. #2
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    Re: Divide a triangle into 2 parts equal area by a line parallel to its base.

    Hello, Jammix!

    Divide a triangle in two equal areas by the line parallel to its base.
    Code:
        -       *
        :      *: *
        :     * :h  *
        H    *  :     *
        :   * * * * * * *
        :  *      b       *
        - * * * * * * * * * *
                   B
    The large triangle has base B and height H.
    The small triangle has base b and height h.

    We want: . \tfrac{1}{2}bh \:=\:\tfrac{1}{2}\left(\tfrac{1}{2}BH\right) \quad\Rightarrow\quad bh \:=\:\tfrac{1}{2}BH\;\;[1]

    From similar triangles: . \frac{b}{h} \,=\,\frac{B}{H} \quad\Rightarrow\quad b \,=\,\tfrac{B}{H}h

    Substitute into [1]: . \left(\tfrac{B}{H}h\right)h \:=\:\tfrac{1}{2}BH \quad\Rightarrow\quad h^2 \:=\:\tfrac{1}{2}H^2

    Therefore: . h \:=\:\frac{H}{\sqrt{2}}
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