Multiple choice

Let x be the area of a square inscribed in a circle of radius r and y be the area of an equilateral triangle inscribed in the same circle. Which one of the following is correct?

  1. 9x2 = 16y2

  2. 27x2 = 64y2

  3. 36x2 = 49y2

  4. 16x2 = 21y2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Square in circle of radius r: diagonal = 2r, side = sqrt(2)r, area x = 2r^2. Equilateral triangle in circle of radius r: side = sqrt(3)r, area y = (sqrt(3)/4) * (sqrt(3)r)^2 = (3*sqrt(3)/4)r^2. x^2 = 4r^4. y^2 = (27/16)r^4. x^2/y^2 = 4 / (27/16) = 64/27. 27x^2 = 64y^2.

AI explanation

The diagonal of a square inscribed in a circle of radius r equals the diameter 2r, making its side length r * sqrt(2) and its area x = 2r^2. For an equilateral triangle inscribed in the same circle, the side length is r * sqrt(3), making its area y = (3 * sqrt(3) / 4) * r^2. Taking the ratio x/y gives 2 / (3 * sqrt(3) / 4), and squaring this ratio yields x^2 / y^2 = 64 / 27, which rearranges to 27x^2 = 64y^2. The result is 27x2 = 64y2.

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