Question

a regular hexagon 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 hexagon.

which transformation(s) must map the hexagon exactly onto itself? choose all that apply.

• clockwise rotation about ( p ) by ( 300^circ )
• reflection across line ( a )
• reflection across line ( b )
• counterclockwise rotation about ( p ) by ( 144^circ )
• none of the above

Explanation:

1. Clockwise rotation about \( P \) by \( 300^\circ \): A regular hexagon has rotational symmetry of order 6, meaning the smallest angle of rotation that maps it onto itself is \( \frac{360^\circ}{6} = 60^\circ \). A \( 300^\circ \) clockwise rotation is equivalent to a \( 360^\circ - 300^\circ = 60^\circ \) counterclockwise rotation, which is a valid rotational symmetry. So this rotation maps the hexagon onto itself.
2. Reflection across line \( a \): Line \( a \) bisects each side it passes through. In a regular hexagon, such a line (a line of symmetry through the midpoints of opposite sides) will reflect the hexagon onto itself, as the hexagon is symmetric with respect to this line.
3. Reflection across line \( b \): Line \( b \) passes through a vertex and bisects a side. In a regular hexagon, a line through a vertex and the midpoint of the opposite side is also a line of symmetry, so reflecting across line \( b \) will map the hexagon onto itself.
4. Counterclockwise rotation about \( P \) by \( 144^\circ \): The rotational symmetry angles for a regular hexagon are multiples of \( 60^\circ \) (\( 60^\circ, 120^\circ, 180^\circ, 240^\circ, 300^\circ \)). \( 144^\circ \) is not a multiple of \( 60^\circ \), so this rotation does not map the hexagon onto itself.

Answer:

• Clockwise rotation about \( P \) by \( 300^\circ \)
• Reflection across line \( a \)
• Reflection across line \( b \)