Multiple choice

The area of a circle inscribed in an equilateral triangle is $154\ cm^{2}$. The side of triangle is:

  1. $7\sqrt {3}$
  2. $14\sqrt {3}$
  3. $5\sqrt {3}$
  4. $10\sqrt {2}$
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B Correct answer
Explanation

Area of inscribed circle = pi * r^2 = 154, so r^2 = 49, r = 7. For an equilateral triangle, r = side / (2 * sqrt(3)). Thus, side = 2 * sqrt(3) * 7 = 14 * sqrt(3).

AI explanation

The area of the inscribed circle is 154 cm^2, so using the formula Area = pi * r^2 with pi approximated as 22/7 gives r^2 = 154 * 7/22 = 49, making the inradius r = 7 cm. For an equilateral triangle with side length a, the formula for the inradius is r = a * sqrt(3) / 6. Setting 7 = a * sqrt(3) / 6 gives a = 42 / sqrt(3), which rationalizes to 14 * sqrt(3) cm.

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