Show that apothem of the equilateral triangle, of which one side length is a, will be sqrt3/6a?

Show that apothem of the equilateral triangle, of which one side length is a, will be $\frac{\sqrt{3}}{6} a$.

Dec 11, 2015

$\Delta O A B$ is isosceles.

Explanation:

$\Delta A B C$ is equilateral. $O H$ is the apothem. $H \in A B$.

In Delta OHA, hat O = (120º)/2 = 60º, hat H = 90º

|OH| = x ; |OA| = r, |AH| = a/2

tan 60º = frac{|AH|}{|OH|} = (sqrt 3 / 2) / (1 / 2) = (a/2) /x

$\sqrt{3} = \frac{a}{2 x}$

$x = \frac{a}{2 \sqrt{3}}$