Multiple choice

If the distance from the vertex to the centroid of an equilateral triangle is $6$ cm, then what is the area of the triangle$?$

  1. $24$ cm$^{2}$
  2. $27\sqrt{3}$ cm$^{2}$
  3. $12$ cm$^{2}$
  4. $12\sqrt{3}$ cm$^{2}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In an equilateral triangle, the distance from vertex to centroid is 2/3 of the height (h). 2/3 * h = 6, so h = 9. Area = (h^2)/sqrt(3) = 81/sqrt(3) = 27*sqrt(3).

AI explanation

In an equilateral triangle, the distance from the vertex to the centroid equals two-thirds of the altitude, making the total altitude 9 cm. The side length (s) is found using s = 2h / sqrt(3), which gives 18 / sqrt(3) = 6 * sqrt(3) cm. The area formula (sqrt(3)/4 * s^2) yields sqrt(3)/4 * 108 = 27 * sqrt(3) cm^2.

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