Question # 4 of 13 A FLAG QUESTION How many distinct values of R, the angle of rotation, are there where 0° < R< 360, that will map a regular pentagon onto itself? Assume that the center of rotation is the center of the pentagon? Answers A - D А З В 4 с 5 D 6

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### Understanding Rotational Symmetry in a Regular Pentagon

**Question #4 of 13:**

How many distinct values of R, the angle of rotation, are there where \( 0^\circ < R < 360^\circ \), that will map a regular pentagon onto itself? Assume that the center of rotation is the center of the pentagon.

#### Answer Choices:
- **A.** 3
- **B.** 4
- **C.** 5
- **D.** 6

### Explanation:

In geometry, the concept of rotational symmetry refers to a figure's ability to look the same after some rotation by a partial turn. For a regular pentagon (a five-sided polygon with equal sides and angles), the points of symmetry occur when the pentagon maps onto itself through rotations.

To determine the distinct angles of rotation \( R \), consider the following:

1. **Total Rotation:** A full circle is 360 degrees.
2. **Number of Sides:** A regular pentagon has 5 sides.
3. **Calculating Rotation Angles:** The rotation that maps the pentagon onto itself is \( \frac{360^\circ}{5} \). This results in a basic angle of 72 degrees.

The possible rotations are then multiples of 72 degrees within the range \( 0^\circ < R < 360^\circ \):
- 72 degrees
- 144 degrees
- 216 degrees
- 288 degrees

Since 360 degrees is not strictly less than 360 degrees, it’s not included. Therefore, there are 4 distinct values for \( R \).

**Correct Answer: B. 4**
Transcribed Image Text:### Understanding Rotational Symmetry in a Regular Pentagon **Question #4 of 13:** How many distinct values of R, the angle of rotation, are there where \( 0^\circ < R < 360^\circ \), that will map a regular pentagon onto itself? Assume that the center of rotation is the center of the pentagon. #### Answer Choices: - **A.** 3 - **B.** 4 - **C.** 5 - **D.** 6 ### Explanation: In geometry, the concept of rotational symmetry refers to a figure's ability to look the same after some rotation by a partial turn. For a regular pentagon (a five-sided polygon with equal sides and angles), the points of symmetry occur when the pentagon maps onto itself through rotations. To determine the distinct angles of rotation \( R \), consider the following: 1. **Total Rotation:** A full circle is 360 degrees. 2. **Number of Sides:** A regular pentagon has 5 sides. 3. **Calculating Rotation Angles:** The rotation that maps the pentagon onto itself is \( \frac{360^\circ}{5} \). This results in a basic angle of 72 degrees. The possible rotations are then multiples of 72 degrees within the range \( 0^\circ < R < 360^\circ \): - 72 degrees - 144 degrees - 216 degrees - 288 degrees Since 360 degrees is not strictly less than 360 degrees, it’s not included. Therefore, there are 4 distinct values for \( R \). **Correct Answer: B. 4**
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