Question

transformations
identifying transformations that map a regular polygon onto itself
a regular pentagon is shown below.
line a bisects each side it passes through.
line b passes through a vertex and bisects a side.
point p is the center of the pentagon.
which transformation(s) must map the pentagon exactly onto itself? choose all that apply.
reflection across line b
counterclockwise rotation about p by 144°
clockwise rotation about p by 288°
reflection across line a
none of the above

Explanation:

Step1: Analyze Reflection across line \( b \)

A regular pentagon has 5 lines of symmetry. Line \( b \) passes through a vertex and bisects the opposite side, so it is a line of symmetry. Reflecting across line \( b \) will map the pentagon onto itself.

Step2: Analyze Counterclockwise rotation about \( P \) by \( 144^\circ \)

The central angle between two adjacent vertices of a regular pentagon is \( \frac{360^\circ}{5}=72^\circ \). A rotation by \( 144^\circ = 2\times72^\circ \) will map a vertex to another vertex, and since the pentagon is regular, this rotation maps the pentagon onto itself.

Step3: Analyze Clockwise rotation about \( P \) by \( 288^\circ \)

A clockwise rotation by \( 288^\circ \) is equivalent to a counterclockwise rotation by \( 360^\circ - 288^\circ = 72^\circ \times 4 \). Since \( 72^\circ \) is the central angle between adjacent vertices, rotating by \( 288^\circ \) (or \( 72^\circ\times4 \)) will map the pentagon onto itself.

Step4: Analyze Reflection across line \( a \)

Line \( a \) bisects each side it passes through, which means it is also a line of symmetry (since in a regular pentagon, such a line will reflect the pentagon onto itself). So reflecting across line \( a \) maps the pentagon onto itself.

Answer:

Reflection across line \( b \), Counterclockwise rotation about \( P \) by \( 144^\circ \), Clockwise rotation about \( P \) by \( 288^\circ \), Reflection across line \( a \) (i.e., all the options except "None of the above" are correct as per the analysis of each transformation for a regular pentagon's symmetry).