Triangle ABC is an equilateral triangle and O is the center of its inscribed circle. If the area of the circle is 16*pi sq cm, what is the area, in square centimeters, of triangle ABC? Express your answer in simplest radical form.

Guest Dec 17, 2021

#1**0 **

1) Find the radius of the inscribed circle by using the formula: Area = pi·r^{2}.

2) Find the length of each side of the equilateral triangle by using the formula:

radius of the incircle = side of the equilateral triangle / 3

3) Find the area of the equilateral triangle by using the formula: Area = ( sqrt(3) / 4 ) · side of equilateral triangle

geno3141 Dec 17, 2021

#2**0 **

Area of circle = pi(r^2) = 16pi => r = 4 cm.

ABC is equilateral, A = B = C = 60^{0}.

Consider circle center O inscribed in ABC: from center O drop a perpendicular, which is the radius r to base of triangle BC at M

which is the midpoint of BC and from O joined to B. This forms another right triangle OBM where B = 30^{0}, O = 60^{0} and M = 90^{0}.

OM = r = 4 cm, BM = 4/(tan 30^{0}) = 4/(1/√3) = 4√3.

Length of BC = 2(BM) = 8√3.

Area of ABC = (1/2)(8√3)(8√3)(sin 60^{0}) = (1/2)(8√3)(8√3)(√3/2) = (192√3)/4 = 48√3 sq cm.

Guest Dec 17, 2021