Multiple choice

$\displaystyle \Delta ABC$ is an equilateral $\displaystyle \Delta $ with area $\displaystyle 36\sqrt{3}cm^{2}$ The area of the inscribed circle is

  1. $\displaystyle 12\pi \: cm^{2}$
  2. $\displaystyle 48\pi \: cm^{2}$
  3. $\displaystyle 24\pi \: cm^{2}$
  4. $\displaystyle 36\pi \: cm^{2}$
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A Correct answer
Explanation

Area of equilateral triangle = (sqrt(3)/4) * side^2 = 36*sqrt(3). side^2 = 144, side = 12. Inradius r = side / (2*sqrt(3)) = 12 / (2*sqrt(3)) = 6/sqrt(3) = 2*sqrt(3). Area of inscribed circle = pi * r^2 = pi * (2*sqrt(3))^2 = pi * 12 = 12*pi.

AI explanation

Using the formula for the area of an equilateral triangle, Area = (side^2 * sqrt(3)) / 4, we equate 36*sqrt(3) to (a^2 * sqrt(3)) / 4 to find the side length a = 12. The radius of the incircle is given by r = a / (2 * sqrt(3)), which is 12 / (2 * sqrt(3)) = 2 * sqrt(3). The area of the incircle is pi * r^2, which calculates to pi * (2 * sqrt(3))^2 = 12*pi. The required area is 12*pi cm^2.

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