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Difficulty: Hard

A square is inscribed in a circle. The radius of the circle is $StartFraction 20 StartRoot 2 EndRoot Over 2 EndFraction$ inches. What is the side length, in inches, of the square?

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Explanation

Choice A is correct. When a square is inscribed in a circle, a diagonal of the square is a diameter of the circle. It's given that a square is inscribed in a circle and the length of a radius of the circle is $StartFraction 20 StartRoot 2 EndRoot Over 2 EndFraction$ inches. Therefore, the length of a diameter of the circle is $2 left parenthesis StartFraction 20 StartRoot 2 EndRoot Over 2 EndFraction right parenthesis$ inches, or $20 StartRoot 2 EndRoot$ inches. It follows that the length of a diagonal of the square is $20 StartRoot 2 EndRoot$ 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 $20 StartRoot 2 EndRoot$ 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 $StartRoot 2 EndRoot$ 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 $20 StartRoot 2 EndRoot equals StartRoot 2 EndRoot left parenthesis s right parenthesis$ . Dividing both sides of this equation by $StartRoot 2 EndRoot$ yields $20 equals s$ . 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.