Subjects geometry

Regular Hexagon 761223

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1. **State the problem:** Determine if the hexagon JNLKOM constructed by Veronica is a regular hexagon inscribed in circle C. 2. **Recall definitions and properties:** - A regular hexagon inscribed in a circle has all sides equal and all vertices lie on the circle. - The diameter JK is a straight line through the center of circle C. - The chord LM is the perpendicular bisector of JK, so it passes through the center and is perpendicular to JK. - Radii CN and CO bisect vertical angles JCL and KCM, meaning points N and O lie on the circle and are positioned symmetrically. 3. **Analyze the construction:** - Since JK is a diameter, the center C is midpoint of JK. - LM is perpendicular bisector of JK, so L and M lie on the circle and are symmetric about C. - Radii CN and CO bisect vertical angles at C, so points N and O lie on the circle and are positioned to create equal arcs. 4. **Conclusion:** - The points J, N, L, K, O, M all lie on circle C. - The construction ensures equal arcs between these points, forming equal sides. - Therefore, hexagon JNLKOM is a regular hexagon inscribed in circle C. **Final answer:** A. Veronica is correct: JNLKOM is a regular hexagon inscribed in circle C.