Question

1 which set of instructions should result in a square inscribed in a circle?
○ start with a circle with center c. draw a chord and label the endpoints p and q. rotate chord (overline{pq}) by (90^circ) about point c. rotate the image (90^circ) around point c, and then rotate the image of the segment (90^circ) about c.
○ start with a circle with center c. draw a radius and label the endpoint p. rotate (overline{pc}) by (45^circ) about point c. then rotate the image (45^circ) about point c. continue rotating each image by (45^circ) about point c until you reach (overline{pc}). connect adjacent intersection points on the circle.
○ start with a circle with center c. draw a diameter. construct a perpendicular to the diameter through c that intersects the circle. connect adjacent intersection points on the circle with segments.
● start with a circle with center c. draw a diameter and label the endpoints p and q. construct a perpendicular to the diameter through c that intersects the circle at points r and s. construct the tangents to the circle at points p, q, r, and s. connect the adjacent intersection points of these tangents.

Explanation:

To inscribe a square in a circle, we need to create four equal - length chords that are perpendicular to each other (since the interior angles of a square are 90° and all sides are equal, and the center of the circle is equidistant from all vertices of the square).

• For the first option: Rotating a chord by 90° multiple times will not necessarily create a square. The length of the chord and the rotation may not result in four sides of equal length and right angles in the correct configuration.
• For the second option: Rotating a radius by 45° each time will create angles that are not 90° between adjacent sides, so it will not form a square.
• For the third option:
• First, we start with a circle and a diameter. Constructing a perpendicular to the diameter through the center (C) gives us another diameter that is perpendicular to the first one.
• The intersection points of these diameters with the circle are four points on the circle.
• When we connect adjacent intersection points, we are connecting points that are equidistant from the center (since they are on the circle) and the angle between the radii to adjacent points is 90° (because the diameters are perpendicular). So, the quadrilateral formed has four equal - length sides (all radii of the circle) and four right angles, which is a square.
• For the fourth option: The construction involves tangents, which are not part of the inscribed square (an inscribed square has its vertices on the circle, while tangents touch the circle at one point and are outside the circle - related construction for the inscribed square).

Answer:

The third option (Start with a circle with center C. Draw a diameter. Construct a perpendicular to the diameter through C that intersects the circle. Connect adjacent intersection points on the circle with segments.)