Multiple choice

A circle is inscribed in a square and then a smaller square is inscribed in the circle. The ratio of the area of the smaller square to that of the larger square is

  1. $1:4$
  2. $\sqrt{2} :2$
  3. $1:2$
  4. $1:\sqrt{2}$
  5. $2:3$
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C Correct answer
Explanation

Let the larger square have side 2s. The inscribed circle has diameter 2s, so radius s. The smaller square inscribed in this circle has a diagonal equal to the circle's diameter (2s). Side of smaller square = 2s / sqrt(2) = s * sqrt(2). Area of larger square = (2s)^2 = 4s^2. Area of smaller square = (s * sqrt(2))^2 = 2s^2. Ratio = 2s^2 / 4s^2 = 1:2.

AI explanation

Let the side length of the larger square be x, meaning the inscribed circle has a diameter of x and a radius of x/2. The diagonal of the smaller inscribed square equals the diameter of the circle, so the side length of the smaller square is x/sqrt(2). The area of the smaller square is (x/sqrt(2))^2 = x^2/2, while the area of the larger square is x^2. The ratio of the smaller square's area to the larger square's area is (x^2/2) divided by x^2, which is 1:2.

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