To construct an equilateral triangle inscribed in a circle, Chase needs to ensure the vertices of the triangle are equidistant from each other and lie on the circle. Here’s the reasoning for the correct answer:

1. Chase already drew a diameter, DE, of the circle.
2. The remaining two vertices of the triangle, F and G, must be positioned such that all sides of the triangle (EF, FG, GE) are equal.

For this to occur:

• Points F and G must be located symmetrically around DE, and they must lie on the circle.
• This symmetry is achieved when the chord connecting F and G is perpendicular to and bisects the diameter DE (since a perpendicular bisector ensures equal distances).

Thus, the correct answer is:

A. Points F and G are endpoints of a chord that is perpendicular to and bisects DE.

This ensures the equilateral triangle is correctly inscribed within the circle.

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Would you like a detailed step-by-step explanation for constructing the triangle or clarification on any other part?

Here are five related questions for deeper understanding:

1. Why does the chord need to be perpendicular to the diameter for symmetry in this construction?
2. How does the perpendicular bisector guarantee that the triangle is equilateral?
3. What is the geometric significance of inscribing a triangle in a circle?
4. How would the process differ if Chase wanted to construct a square instead of a triangle?
5. Could Chase use any other method to find F and G, and if so, how?

Tip: To construct shapes inscribed in a circle, always consider the circle's symmetry and properties like equal angles and equal arc lengths.