Question

given the regular polygon, select all of the rotations and reflections that carry the figure onto itself. a a rotation of 72° around the center point d b a reflection across line a, through two vertices c a reflection across line b, through one vertex, the center d d a rotation of 60° around the center point d e a reflection across line c, and the midpoint of the opposite side

Explanation:

Step1: Recall properties of regular pentagon

A regular pentagon has rotational symmetry. The central angle for rotation is $\frac{360^{\circ}}{n}$, where $n = 5$ (number of sides). So, $\frac{360^{\circ}}{5}=72^{\circ}$. A rotation of $72^{\circ}$ around the center point $d$ will map the pentagon onto itself.

Step2: Recall reflection properties

A regular pentagon has reflection - symmetry. Lines of symmetry can be drawn through a vertex and the mid - point of the opposite side. Line $b$ is a line of symmetry passing through a vertex and the center $d$, and a reflection across line $b$ will map the pentagon onto itself. Also, line $c$ is a line of symmetry passing through the mid - point of a side and the mid - point of the opposite side, and a reflection across line $c$ will map the pentagon onto itself.

Step3: Analyze rotation of 60°

Since $\frac{360^{\circ}}{5}=72^{\circ}$, a rotation of $60^{\circ}$ is not a rotational symmetry angle for a regular pentagon, so it will not map the pentagon onto itself.

Step4: Analyze reflection across line a

Line $a$ does not pass through a vertex and the mid - point of the opposite side or through the mid - points of opposite sides in a way that would be a line of symmetry for the pentagon, so a reflection across line $a$ will not map the pentagon onto itself.

Answer:

A. a rotation of 72° around the center point d
C. a reflection across line b, through one vertex, the center d
E. a reflection across line c, and the midpoint of the opposite side