Question

which of the following is true of the constructions of an equilateral triangle, a square, and a regular hexagon when they are inscribed in circles? the diameter in the first step of the constructions divides each shape in half. three diameters are needed to construct each inscribed polygon. the arc in step two of each construction divides the circle in half. the line segments that form each shape are congruent to the diameter of the circle.

Explanation:

1. Recall the properties of in - scribed polygons:

• For an equilateral triangle, square, and regular hexagon inscribed in a circle, a diameter of the circle is a line of symmetry for each of these symmetric shapes. When we start the construction and consider a diameter of the circle, it divides each of these inscribed shapes into two congruent halves due to their symmetry about the diameter.
• Analyze each option:
• Option 1: A diameter is a line of symmetry for an equilateral triangle, square, and regular hexagon inscribed in a circle, so it divides each shape in half. This is correct.
• Option 2: For an equilateral triangle, we don't need three diameters for construction. We can use a compass to construct it with one diameter as a starting - point. For a square, we can use two perpendicular diameters. For a regular hexagon, we can use one diameter and then use the radius to mark off the vertices. So this option is incorrect.
• Option 3: The arcs in the construction steps are used to mark off vertices of the polygons, not to divide the circle in half in the general sense of construction steps for these polygons. So this option is incorrect.
• Option 4: In an equilateral triangle inscribed in a circle, the side - length is $\sqrt{3}r$ (where $r$ is the radius of the circle), for a square it is $\sqrt{2}r$, and for a regular hexagon it is $r$. None of these are congruent to the diameter ($2r$) of the circle. So this option is incorrect.

Answer:

The diameter in the first step of the constructions divides each shape in half.