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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.

 Dec 17, 2021
 #1
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1)  Find the radius of the inscribed circle by using the formula:  Area = pi·r2.

 

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

 Dec 17, 2021
 #2
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Area of circle = pi(r^2) = 16pi => r = 4 cm.

ABC is equilateral, A = B = C = 600.

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 = 300, O = 600 and M = 900.

OM = r = 4 cm, BM = 4/(tan 300) = 4/(1/√3) = 4√3. 

Length of BC = 2(BM) = 8√3.

Area of ABC = (1/2)(8√3)(8√3)(sin 600) = (1/2)(8√3)(8√3)(√3/2) = (192√3)/4 = 48√3 sq cm.

 Dec 17, 2021

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