Splitting a 30, 60, 90 degree triangle's area into two equal halves.

Question

!(https://i.stack.imgur.com/vrfl3.jpg) According to the photo, what would the value of 'h' be in relation to 'W'? (e.g.: h=(1/3)W)

How would you find that out?

(The middle line (parallel to the base) splits the triangle's area perfectly in half)

Answer 1

The triangles are similar because they have the $30$ and $90$ degree angles in common. The area goes as the square of the sides, so the smaller sides are $\frac 1{\sqrt 2}$ of the corresponding large sides. That means $h=(1-\frac 1{\sqrt 2})W$.. This does not depend on the triangle being $30-60-90$, just the similarity, so any triangle cut by a line parallel to a side gives the same result.