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?

1 Answer
Dec 25, 2015

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)