Multiple choice

A circle is inscribed in an equilateral triangle of side 24 cm. What is the area (in cm2) of a square inscribed in the circle?

  1. 48

  2. 72

  3. 54

  4. 96
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D Correct answer
Explanation

Inradius of equilateral triangle = side / (2 * sqrt(3)) = 24 / (2 * sqrt(3)) = 4 * sqrt(3). The circle radius r = 4 * sqrt(3). A square inscribed in a circle has diagonal = 2r = 8 * sqrt(3). Area of square = (diagonal^2) / 2 = (64 * 3) / 2 = 96.

AI explanation

For an equilateral triangle, the radius of the inscribed circle is the height of the triangle divided by 3, and the height is found by multiplying the side length by the square root of 3 divided by 2. With a side length of 24 cm, the height is 12 times the square root of 3, making the radius 4 times the square root of 3 cm. This radius acts as the circumradius for the inscribed square, and the diagonal of such a square equals 2 times the radius, which is 8 times the square root of 3. The area of the square is half the square of its diagonal, calculated as one half of 192, which is 96 square centimeters.

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