SOLUTION: the area of an equilateral triangle is 108 square root of 3 feet squared what is the length of a side and the apothem in simplest radical form?

Question 969603: the area of an equilateral triangle is 108 square root of 3 feet squared what is the length of a side and the apothem in simplest radical form?
Answer by Boreal(15235) (Show Source): You can put this solution on YOUR website!
Side=12 sqrt (3) feet; ap=6 feet ANSWER
A=1/2 * bh
108 sqrt (3) sq ft=(1/2) bh

Equilateral triangles with side s have an altitude of (s/2) sqrt (3). The altitude splits the base into 2, so the triangle formed with the altitude is a 30-60-90 triangle with the hypotenuse s, the altitude is (s/2)sqrt(3).
Area is (1/2) *s *(s/2)*sqrt(3)=[(s^2)/4]sqrt (3)=108 sqrt (3)
The sqrt(3) on both sides divide out,
108=(s^2/4)
Multiply by 4
432= s^2
s= sqrt (432) In simple radical form would be sqrt (144*3)=12 sqrt(3) feet
The apothem is 1/3 the altitude. or (1/3)*(s/2)sqrt(3)=(s/6)sqrt(3)
But s=12 sqrt (3) so (s/6) sqrt (3)=12/6 (sqrt 3)^2=2*3 =6 feet
sqrt 3^2=3

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