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

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

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Answer 1 · 2015-12-11T16:47:15

#Delta OAB# is isosceles.

#Delta ABC# is equilateral. #OH# is the apothem. #H in AB#.

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 = a / (2x)#

#x = a / (2 sqrt 3)#