Problem

\(\text{Six}\) points \(P_{0}\) through \(P_{5}\) are equally spaced in order on a circle of radius \(R\). A student tries to form an inscribed equilateral triangle by connecting \(P_{0}\), \(P_{1}\), and \(P_{2}\). Which complete response correctly evaluates the student's attempt?

Six equally spaced circle points labeled P0 through P5, with the student's consecutive choices P0, P1, and P2 highlighted for diagnosis.

Answer choices

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invalid action = marking six equal points; correction = use P0,P1,P2; invariant = selected vertices are 120° apart; student's side lengths = √3R,√3R,√3R

invalid action = using points on one circle; correction = move P2 off the circle; invariant = central angles are 120°; student's side lengths = R,R,√3R

invalid action = skipping one point each time; correction = use three consecutive points; invariant = all chords have length R; student's side lengths = R,R,R

invalid action = selecting three consecutive points P0,P1,P2; correction = connect every other point, such as P0,P2,P4; invariant = successive selected vertices are 120° apart and all three chords have length √3R; student's side lengths = R,R,√3R

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\(\text{Six}\) points \(P_{0}\) through \(P_{5}\) are equally spaced in order on a circle of radius \(R\). A student tries to form an inscribed equilateral triangle by connecting \(P_{0}\), \(P_{1}\), and \(P_{2}\). Which complete response correctly evaluates the student's attempt?

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