Question

question 3 points 2 which of the following transformation carry this regular pentagon onto itself? image of regular pentagon with dashed line m rotation of 72° counterclockwise rotation of 30° clockwise rotation of 45° counterclockwise reflection across m

Explanation:

Step1: Analyze rotational symmetry

A regular pentagon has a rotational symmetry order of 5. The central angle for each rotation that maps it onto itself is $ \frac{360^\circ}{5} = 72^\circ $. So a rotation of $ 72^\circ $ (clockwise or counterclockwise) will carry it onto itself.

Step2: Analyze reflection symmetry

For a regular pentagon, the reflection lines pass through a vertex and the midpoint of the opposite side. The line \( M \) in the diagram connects two vertices (since it's a dashed line through two vertices of the pentagon), so reflecting across \( M \) (a line of symmetry) will also carry the pentagon onto itself. Wait, but let's check the options again. Wait, the options: rotation of \( 72^\circ \) counterclockwise is valid, and reflection across \( M \) – but wait, in a regular pentagon, the reflection axis through two vertices (the line \( M \) here connects two vertices) is a line of symmetry. Wait, but let's re - evaluate.

Wait, first, rotational symmetry: the angle of rotation for a regular \( n \) - gon is \( \frac{360^\circ}{n} \). For \( n = 5 \), it's \( 72^\circ \). So a rotation of \( 72^\circ \) counterclockwise will map the pentagon onto itself. Now, for reflection: in a regular pentagon, the reflection lines are either through a vertex and the mid - point of the opposite side or through two vertices? Wait, no, in a regular pentagon, each reflection axis passes through one vertex and the mid - point of the opposite edge. Wait, the line \( M \) in the diagram: looking at the pentagon, the dashed line \( M \) connects two vertices. Wait, maybe my initial thought was wrong. Wait, let's count the number of sides. A regular pentagon has 5 sides. The angle between two adjacent vertices from the center is \( 72^\circ \).

Wait, let's check the rotation option first. The rotation of \( 72^\circ \) counterclockwise: since \( \frac{360}{5}=72 \), rotating by \( 72^\circ \) will move each vertex to the position of the next vertex, so the pentagon maps onto itself. Now, the reflection across \( M \): if \( M \) is a line that does not pass through a vertex and the mid - point of the opposite edge, but through two vertices, is that a line of symmetry? Wait, no, in a regular pentagon, the reflection axes are through a vertex and the mid - point of the opposite side. So maybe the line \( M \) is not a line of symmetry. Wait, maybe I made a mistake. Let's re - examine the problem.

Wait, the options are:

1. rotation of \( 72^\circ \) counterclockwise

2. rotation of \( 30^\circ \) clockwise

3. rotation of \( 45^\circ \) counterclockwise

4. reflection across \( M \)

First, for rotation: \( 30^\circ \) and \( 45^\circ \) are not multiples of \( 72^\circ \), so they won't map the pentagon onto itself. For \( 72^\circ \) rotation, as \( \frac{360}{5}=72 \), rotating by \( 72^\circ \) counterclockwise will make each vertex coincide with the next vertex, so the pentagon maps onto itself. Now, for reflection across \( M \): if \( M \) is a line that is not a line of symmetry (since in a regular pentagon, lines of symmetry pass through a vertex and the mid - point of the opposite edge, not through two vertices), then reflection across \( M \) will not map the pentagon onto itself. Wait, but maybe the diagram shows that \( M \) is a line of symmetry. Wait, the pentagon in the diagram: the dashed line \( M \) connects two vertices. Let's think again. A regular pentagon has 5 lines of symmetry, each passing through a vertex and the mid - point of the opposite side. If the line \( M \) connects two vertices, then it's not a line of symmetry. So the correct transformation is the rotation of \( 72^\circ \) counterclockwise.

Answer:

rotation of \( 72^\circ \) counterclockwise