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9.3.35 Point O is the center of the regular nonagon shown at the right. A B. Find the angle of rotation about O that maps D to I. H A rotation about O maps D to I.

9.3.35 Point O is the center of the regular nonagon shown at the right. A B. Find the angle of rotation about O that maps D to I. H A rotation about O maps D to I.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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### Understanding Rotation Angles in Regular Polygons #### Problem 9.3.35: **Prompt:** Point O is the center of the regular nonagon shown at the right. Find the angle of rotation about O that maps D to I. **Solution Space:** A _____° rotation about O maps D to I. **Answer Submission:** Enter your answer in the answer box and then click Check Answer. --- #### Diagram Explanation: The diagram consists of a regular nonagon (a nine-sided polygon) with the center labeled as point O. The vertices of the nonagon are labeled in a clockwise manner starting from vertex A. The complete sequence is: - A - B - C - D - E - F - G - H - I #### Calculating the Angle of Rotation: To find the angle of rotation that maps vertex D to vertex I, we need to determine the number of vertices between D and I and the angle subtended per vertex in the nonagon. 1. **Total Internal Angles in Nonagon:** - A regular nonagon has 9 sides. - The total internal angles of a polygon = (n - 2) × 180° - For a nonagon: (9 - 2) × 180° = 7 × 180° = 1260° 2. **Angle per Vertex in Regular Nonagon:** - Since it’s a regular polygon, the rotation to map a vertex to the next adjacent vertex is equal. - Each internal angle at the center = 360° / number of sides - For nonagon: 360° / 9 = 40° 3. **Distance from D to I:** - Count the vertices from D to I in a clockwise direction: - D → E → F → G → H → I - Number of steps = 5 4. **Total Rotation Angle:** - Total rotation angle = steps × angle per step - Total rotation angle = 5 × 40° = 200° Hence, a 200° rotation about point O maps point D to point I. **Final Answer:** A **200°** rotation about O maps D to I. --- **Note for Students:** Understanding how to calculate rotation angles in regular polygons is fundamental in geometry. Ensure you are comfortable with dividing the total degrees around a point (360
Transcribed Image Text:### Understanding Rotation Angles in Regular Polygons #### Problem 9.3.35: **Prompt:** Point O is the center of the regular nonagon shown at the right. Find the angle of rotation about O that maps D to I. **Solution Space:** A _____° rotation about O maps D to I. **Answer Submission:** Enter your answer in the answer box and then click Check Answer. --- #### Diagram Explanation: The diagram consists of a regular nonagon (a nine-sided polygon) with the center labeled as point O. The vertices of the nonagon are labeled in a clockwise manner starting from vertex A. The complete sequence is: - A - B - C - D - E - F - G - H - I #### Calculating the Angle of Rotation: To find the angle of rotation that maps vertex D to vertex I, we need to determine the number of vertices between D and I and the angle subtended per vertex in the nonagon. 1. **Total Internal Angles in Nonagon:** - A regular nonagon has 9 sides. - The total internal angles of a polygon = (n - 2) × 180° - For a nonagon: (9 - 2) × 180° = 7 × 180° = 1260° 2. **Angle per Vertex in Regular Nonagon:** - Since it’s a regular polygon, the rotation to map a vertex to the next adjacent vertex is equal. - Each internal angle at the center = 360° / number of sides - For nonagon: 360° / 9 = 40° 3. **Distance from D to I:** - Count the vertices from D to I in a clockwise direction: - D → E → F → G → H → I - Number of steps = 5 4. **Total Rotation Angle:** - Total rotation angle = steps × angle per step - Total rotation angle = 5 × 40° = 200° Hence, a 200° rotation about point O maps point D to point I. **Final Answer:** A **200°** rotation about O maps D to I. --- **Note for Students:** Understanding how to calculate rotation angles in regular polygons is fundamental in geometry. Ensure you are comfortable with dividing the total degrees around a point (360
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