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)#