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A square is inscribed in a circle. The radius of the circle is $\\frac{20\\sqrt{2}}{2}$ inches. What is the side length, in inches, of the square?

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A square is inscribed in a circle. The radius of the circle is $\\frac{20\\sqrt{2}}{2}$ inches. What is the side length, in inches, of the square?

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Choice A is correct. When a square is inscribed in a circle, the diagonal of the square is the diameter of the circle. It's given that a square is inscribed in a circle and the length of the radius of the circle is $\\frac{20\\sqrt{2}}{2}$ inches. Therefore, the length of the diameter of the circle is $2\\left(\\frac{20\\sqrt{2}}{2}\\right)$ inches, or $20\\sqrt{2}$ inches. It follows that the length of a diagonal of the square is $20\\sqrt{2}$ inches. A diagonal of a square separates the square into two right triangles in which the legs are the sides of the square and the hypotenuse is a diagonal. Since a square has 4 congruent sides, each of these two right triangles has congruent legs and a hypotenuse of length $20\\sqrt{2}$ inches. Since each of these two right triangles has congruent legs, they are both $45-45-90$ triangles. In a $45-45-90$ triangle, the length of the hypotenuse is $\\sqrt{2}$ times the length of a leg. Let $s$ represent the length of a leg of one of these $45-45-90$ triangles. It follows that $20\\sqrt{2}=\\sqrt{2}(s)$. Dividing both sides of this equation by $\\sqrt{2}$ yields $20=s$. Therefore, the length of a leg of one of these $45-45-90$ triangles is $20$ inches. Since the legs of these two $45-45-90$ triangles are the sides of the square, it follows that the side length of the square is $20$ inches.

Choice B is incorrect. This is the length of a radius, in inches, of the circle.

Choice C is incorrect. This is the length of a diameter, in inches, of the circle.

Choice D is incorrect and may result from conceptual or calculation errors.

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A square is inscribed in a circle. The radius of the circle is $\\frac{20\\sqrt{2}}{2}$ inches. What is the side length, in inches, of the square?

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$\\frac{20\\sqrt{2}}{2}$

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$20\\sqrt{2}$

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A square is inscribed in a circle. The radius of the circle is $\frac{20\sqrt{2}}{2}$ inches. What is the side length, in inches, of the square?