Question

the diagram below shows a partly completed construction of a square inscribed in ⊙a with a vertex at b. complete the construction.

Explanation:

Step1: Draw perpendicular to BC

Since \( BC \) is a diameter, to construct the square, we first draw a line perpendicular to \( BC \) through the center \( A \). This line will intersect the circle at two more points, say \( D \) and \( E \).

Step2: Connect the vertices

Now, connect the points \( B \), \( D \), \( C \), \( E \) (in order) to form the square. The key is that the central angles for a square inscribed in a circle are \( 90^\circ \), so the perpendicular to the diameter (which is a straight angle \( 180^\circ \)) will create the necessary \( 90^\circ \) angles between the sides.

Answer:

1. Draw a line perpendicular to \( \overleftrightarrow{BC} \) through center \( A \), intersecting \( \odot A \) at \( D \) and \( E \).
2. Connect \( B \), \( D \), \( C \), \( E \) (and back to \( B \)) to form the square.