A square is inscribed in a circle. The radius of the circle is 20 2 2 inches. What is the

Question

A square is inscribed in a circle. The radius of the circle is 20 2 2 inches. What is the side length, in inches, of the square?

Accepted Answer

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 20 2 2 inches. Therefore, the length of a diameter of the circle is 2 20 2 2 inches, or 20 2 inches. It follows that the length of a diagonal of the square is 20 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 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 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 2 = 2 s . Dividing both sides of this equation by 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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