Icecream Cone:Semicircle On Top of Equilaterial Triangle

[dtweah]

A semicircle is placed upon one side of equilateral △ABC. (Draw so that vertex B is the Bottom of the Cone). The point halfway along the arc of the semicircle is labeled D. Given that BD = 1, what is the length of each side in the equilateral triangle?

(a)2/3
(b)3/4
(c)√3/2
(d)(3 − √3)/2
(e) √3 − 1

[dumb.doofus]

e) root(3) - 1

Solution:

If a is the side of the equilateral triangle.. then,

root(3)a/2 + a/2 = 1

a = 2/(root(3) + 1)

Multiply numerator and denominator by root(3) -1.. this gives

2(root(3) - 1)/(root(3)^2 - 1^2)

= root(3) - 1