Multiple choice

An equilateral triangle is inscribed in the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-12\mathrm{x}-8\mathrm{y}+4=0$ then the length of the side is

  1. $2$
  2. $6$
  3. $4\sqrt{3}$
  4. $12$
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D Correct answer
Explanation

The circle equation x^2 + y^2 - 12x - 8y + 4 = 0 can be rewritten as (x-6)^2 + (y-4)^2 = 36 + 16 - 4 = 48. The radius r is sqrt(48) = 4*sqrt(3). For an equilateral triangle inscribed in a circle, the side length s is r*sqrt(3). Thus, s = 4*sqrt(3) * sqrt(3) = 4 * 3 = 12.

AI explanation

The center and radius of the given circle are found by completing the square, yielding (6, 4) and 4 times the square root of 3. For an equilateral triangle inscribed in a circle, the side length formula is the radius multiplied by the square root of 3. Multiplying the radius 4 times the square root of 3 by the square root of 3 gives the side length as 12.

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