SOLUTION: Equilateral triangle ABC has altitude AD. Median AE of triangle ABD is drawn. If the area of triangle AEC is {{{ 27* sqrt( 3 ) }}} cm^2, what is the side length AB, in cm?

Algebra ->  Triangles -> SOLUTION: Equilateral triangle ABC has altitude AD. Median AE of triangle ABD is drawn. If the area of triangle AEC is {{{ 27* sqrt( 3 ) }}} cm^2, what is the side length AB, in cm?      Log On


   

Question 1106412: Equilateral triangle ABC has altitude AD. Median AE of triangle ABD is drawn. If the area of triangle AEC is +27%2A+sqrt%28+3+%29+ cm^2, what is the side length AB, in cm?
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!



Let BE = ED = x.  Then
BD = 2x = DC.  Then
BC = 4x = AB

By using the Pythagorean theorem on right triangle ADC,
AD = 2x%2Asqrt%283%29

EC = ED + DC = x + 2x = 3x

Area of triangle AEC = expr%281%2F2%29%2Abase%2Aaltitude%22%22=%22%22expr%281%2F2%29EC%2AAD%22%22=%22%22expr%281%2F2%29%283x%29%282x%2Asqrt%283%29%29%22%22=%22%223x%5E2%2Asqrt%283%29

Since the area of triangle AEC = 27%2Asqrt%283%29, we have
the equation:

3x%5E2%2Asqrt%283%29%22%22=%22%2227%2Asqrt%283%29

Divide both sides by 3sqrt%283%29

x%5E2%22%22=%22%229

x%22%22=%22%223

So AB = 4x = 4(3) = 12

Edwin