Question

dylan constructed this figure using a compass with its width set equal to ax, the radius of the circle.
he claims that the measures of the three sides of triangle abx are all equal to ax, making $\triangle abx$ equilateral. since this makes central angle axb measure $60^\circ$, $m\overarc{ab} = 60^\circ$. dylan also claims that by repeatedly applying the same argument, he can prove that the inscribed hexagon is regular.
which statement is true?
a. dylan’s reasoning about arc ab is correct, and the hexagon is regular.
b. dylan’s reasoning about arc ab is correct, but the hexagon is not regular.
c. dylan’s reasoning about arc ab is incorrect, but the hexagon is regular.
d. dylan’s reasoning about arc ab is incorrect, and the hexagon is not regular.
e. dylan’s reasoning about arc ab is correct, but it is not enough information to prove whether the hexagon is regular.

Explanation:

Step1: Verify ΔABX is equilateral

AX = BX = AB (all equal to circle radius), so ΔABX is equilateral.

Step2: Find central angle ∠AXB

For equilateral ΔABX, $m\angle AXB = 60^\circ$.

Step3: Relate central angle to arc measure

The measure of arc $\overset{\frown}{AB}$ equals its central angle, so $m\overset{\frown}{AB}=60^\circ$.

Step4: Prove hexagon is regular

Repeating this, all 6 arcs are $60^\circ$, so all sides (chords of equal arcs) are congruent, and all internal angles of the hexagon are equal. Thus the hexagon is regular.

Answer:

A. Dylan's reasoning about arc AB is correct, and the hexagon is regular.