Final answer:
The vertices of a square inscribed in a circle are located at the midpoint of the sides of the square, as the circle's circumference touches the square exactly at those midpoint locations. d) Anywhere on the square's perimeter
Step-by-step explanation:
The question posed pertains to the position of the vertices of a square in relation to a circle that is inscribed within it. In the context of coordinate geometry, when a circle is inscribed in a square, the sides of the square are tangent to the circle at exactly four points, each located midway along the sides of the square. The vertices of the square, therefore, are outside of the circle and equidistant from the center of the circle along the diagonals of the square. Considering this geometrical arrangement, the correct answer to the question "Then, one of the vertices is:" is c) At the midpoint of a side of the square.
To visualize this, imagine the square and the circle both situated on a coordinate plane where the sides of the square are parallel to the axes. If we take the length of a side of the square to be a, then the diameter of the circle is also a, since it fits perfectly between two opposite sides of the square and the circumference of the circle is smaller than the perimeter of the square (4a) but larger than twice the side of the square (2a). Hence, we can determine that the vertices of the square are not at the center of the circle (a), on the circumference of the circle (b), or anywhere on the square's perimeter (d), but rather at the midpoint of a side of the square.