Question

Provide two different examples of the rotation less than 75 but greater than 0 that will turn a regular dodecagon onto itself

Answer 1

[AI-generated answer — Bot GPT 3.5]

Example 1:

- Rotation of 30 degrees: This will turn the dodecagon by an angle of 30 degrees in a counterclockwise direction. After a full rotation by 360 degrees, the dodecagon will align with itself since 360 divided by 30 equals 12. Therefore, this rotation will preserve the regular dodecagon's symmetry and position.

Example 2:
- Rotation of 60 degrees: This will turn the dodecagon by an angle of 60 degrees in a counterclockwise direction. Similar to the previous example, after a full rotation by 360 degrees, the dodecagon will align with itself since 360 divided by 60 equals 6. Thus, this rotation also preserves the regular dodecagon's symmetry and position.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find rotations that turn a regular dodecagon onto itself, we need to consider the different possible rotations. A regular dodecagon has 12 sides, so we can divide 360 degrees by 12 to determine the measure of each angle. Each angle of a regular dodecagon measures 30 degrees.

Example 1:
To rotate a regular dodecagon onto itself, we can take a single rotation of 30 degrees. This means that each vertex of the dodecagon will be moved 30 degrees counterclockwise to its new position.

Example 2:
We can also consider a rotation of 60 degrees. In this case, each vertex of the dodecagon will be moved 60 degrees counterclockwise to its new position. As 60 degrees is greater than 30 degrees but still less than 75 degrees, it satisfies the given condition.

Answer 3

[AI-generated answer — Explain Bot]

To find the rotation less than 75 but greater than 0 that will turn a regular dodecagon onto itself, we need to determine the angles of rotation that preserve the shape of the dodecagon.

A regular dodecagon has 12 sides, so the total angle around a vertex is 360 degrees (360°). To find the rotation angles that preserve the shape, we divide 360° by the number of sides, which gives us the interior angle of the dodecagon.

360° / 12 = 30°

The first example of a rotation less than 75 but greater than 0 that will turn a regular dodecagon onto itself is a 30° rotation.

Another example is a 60° rotation. This is possible because a 12-sided regular polygon has rotational symmetry, meaning it can be rotated by a multiple of its interior angle and still retain its shape.

To summarize:
- The first example is a 30° rotation that preserves the shape of a regular dodecagon.
- The second example is a 60° rotation that also preserves the shape of a regular dodecagon.

So, a 30° and 60° rotation will turn a regular dodecagon onto itself.