Provide two different degrees of rotation less than 75°
but greater than 0°
that will turn a regular dodecagon onto itself.
A regular dodecagon has 12 sides. In order to turn it onto itself, we need to find angles of rotation that, when applied repeatedly, will align each vertex of the dodecagon onto another existing vertex.
Option 1: 30° Rotation
A 30° rotation will align every second vertex of the dodecagon back onto itself. This means that after applying six consecutive 30° rotations, every vertex will return to its original position.
Option 2: 60° Rotation
A 60° rotation will align every third vertex of the dodecagon back onto itself. This means that after applying four consecutive 60° rotations, every vertex will return to its original position.
Both of these degrees of rotation (30° and 60°) are less than 75° and greater than 0° and will turn a regular dodecagon onto itself.
To find the degrees of rotation that will turn a regular dodecagon onto itself, we can use the formula:
Degree of rotation = 360° / Number of sides
For a regular dodecagon, which has 12 sides, we can calculate the degrees of rotation as follows:
Degree of rotation = 360° / 12 = 30°
So, the two different degrees of rotation less than 75° but greater than 0° that will turn a regular dodecagon onto itself are 30° and 60°.
To find degrees of rotation that will turn a regular dodecagon onto itself, we need to consider the properties of a regular dodecagon.
A regular dodecagon has 12 sides of equal length and 12 angles of equal measure. The sum of the angles in any polygon can be found using the formula (n-2) * 180°, where n is the number of sides. For a regular dodecagon, the sum of the angles is (12-2) * 180° = 1800°.
Since a regular dodecagon has 12 equal angles, each angle measures 1800° / 12 = 150°.
Now, we can find the degrees of rotation that will turn a regular dodecagon onto itself:
1. The smallest rotation would be to rotate the dodecagon by one of its angles, which is 150°. So, rotating the dodecagon by 150° will bring it back to its original position.
2. To find another degree of rotation, we can note that rotating the dodecagon by any multiple of its angles will also bring it back to its original position. Therefore, rotating the dodecagon by 150° * 2 = 300° will also turn it onto itself.
Therefore, two different degrees of rotation less than 75° but greater than 0° that will turn a regular dodecagon onto itself are 150° and 300°.