Question

which figure best shows the construction of point r, where \\( \overline{pr} \\) is one side of an inscribed square in circle \\( o \\)?

Explanation:

To construct an inscribed square in a circle, we first need to find points that are 90° apart (since the central angle for each side of a square inscribed in a circle is 90°). The steps for constructing an inscribed square involve:

1. Drawing a diameter (e.g., \( PQ \) with center \( O \)).
2. Constructing a perpendicular bisector of this diameter (or a line perpendicular to the diameter at one of its endpoints, like \( P \) or \( Q \)) to get another diameter perpendicular to the first, creating four points on the circle (the vertices of the square).

Looking at the figures, the one that shows the construction of a perpendicular line (or the arc markings to find the perpendicular) from the diameter to get the point \( R \) (such that \( \angle POR = 90^\circ \), making \( PR \) a side of the square) is the correct one. Typically, the figure with arc markings (like intersecting arcs to create a perpendicular) or a perpendicular line from the diameter's endpoint (or center - related perpendicular) would be the one. Assuming the middle - left or a figure with arc intersections to create a right angle at the center or on the circle for \( PR \), the figure that has the correct arc - based construction (to get a 90° angle for the square's side) is the answer. If we consider the standard construction: after drawing diameter \( PQ \), we construct a perpendicular at \( P \) (or \( Q \)) to the diameter, which can be done by drawing arcs with the same radius from \( P \) (and another point) to find the perpendicular line, intersecting the circle at \( R \) (and another point). So the figure with the correct arc markings (like two arcs from \( P \) and another point on the circle to create the perpendicular) is the one that best shows the construction of \( R \) for the inscribed square's side \( PR \).

Answer:

(Assuming the second figure from the left, but since the exact labels are a bit unclear, the figure with arc markings to create a perpendicular to the diameter at \( P \) or to form a 90° central angle for \( PR \) is the correct one. If we had to pick from typical options, the figure with arcs intersecting to give a right angle for \( PR \) as a side of the square.)