Question

If you split an equilateral triangle as it sits on its base horizontally, at what height would the top and bottom areas be equal? Using variables?

Answer 1

Let altitude be `h`. Note that a trapezoid and a smaller triangle will occur after you split the triangle. The splitting height (trapezoid's height) is
`color(white)xxh(2-h)`

Explanation:

Let altitude be `h`, one side length be `a`, and area be `A`.
`color(white)xxh/a=sin60`
`color(white)(xxx)=sqrt3/2`

Then one side length of the triangle will be
`color(white)xxa=(2sqrt3h)/3`

`color(white)xxA=hxx2sqrt3/3hxx1/2`
`color(white)(xxx)=sqrt3/3h^2`

`=>A/2=sqrt3/6h^2`

Note that a trapezoid and a smaller triangle will occur after you split the triangle. Suppose that splitting height (trapezoid's height) is `x` and splitting base (trapezoid's upper base) is `b`:

If height of smalll triangle is `h-x`, then:
`color(white)xxh-x=sqrtb/2`

`=>b=2sqrt3/3(h-x)`
`=>A_("Lower")=2sqrt3/3(h-x)`

`color(white)xxA_("Lower")=A/2`
`=>sqrt3/6h^2=sqrt3/3(h-x)`
`=>h^2=2h-2x`

`=>x=h(2-h)`