SOLUTION: Equilateral triangle ABC has altitude AD. Median AE of triangle ABD is drawn. If the area of triangle AEC is 27√3 cm2. What is the side lenght AB, in cm?

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Question 1105588: Equilateral triangle ABC has altitude AD. Median AE of triangle ABD is drawn. If the area of triangle AEC is 27√3 cm2. What is the side lenght AB, in cm?
Found 2 solutions by KMST, greenestamps:
Answer by KMST(5328) About Me  (Show Source):
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An equilateral triangle with side length x has a height %28sqrt%283%29%2F2%29x
,
and an area %28sqrt%283%29%2F4%29x%5E2,
so if x represents the length of side AB in cm, %28sqrt%283%29%2F4%29x%5E2 is the area of ABC in cm%5E2 .

The altitude of an equilateral triangle is also a median,
so it splits the triangle into two triangles with the same area.
Those triangles are ABD and ACD, each with an area equal to {1/2}}} of the area of ABC.
Triangle ABD is similarly split into triangles ABE and AED,
each with area equal to 1%2F2of the area of ABD,
meaning 1%2F4 of the area of ABC.
Triangle ARC is the union of triangles ABD and AED,
so its area is 1%2F2%2B1%2F4=3%2F4 of the area of ABC.
That is
%283%2F4%29%28sqrt%283%29%2F4%29x%5E2=%283sqrt%283%29%2F16%29x%5E2 .

%283sqrt%283%29%2F16%29x%5E2=27sqrt%283%29
%283%2F16%29x%5E2=27
x%5E2=27%2A16%2F3
x%5E2=9%2A16
x=sqrt%289%2A16%29=sqrt%289%29sqrt%2816%29r3%2A4=highlight%2812%29 .
The length of side AB is highlight%2812cm%29 .

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The altitude of an equilateral triangle is also a median. If AE is a median of triangle ABD, then the base of triangle AEC is 3/4 the side length of the equilateral triangle.

If the area of triangle AEC is 27%2Asqrt%283%29, then the area of the equilateral triangle is 4/3 of that, or 36%2Asqrt%283%29.

The area of an equilateral triangle is %28s%5E2%2Asqrt%283%29%29%2F4, so
%28s%5E2%2Asqrt%283%29%29%2F4+=+36%2Asqrt%283%29
s%5E2+=+144
s+=+12

The side length of the equilateral triangle is 12 cm.