Thread: Divide a triangle into 2 parts equal area by a line parallel to its base.

1. 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!!

2. 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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How can we divide a triangle in to two equal areas

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