Question

a regular hexagon is shown below.
line m bisects each side it passes through.
line n passes through a vertex and bisects a side.
point y is the center of the hexagon.
which transformation(s) must map the hexagon exactly onto itself? choose all that apply.
reflection across line m
counterclockwise rotation about y by 288°
clockwise rotation about y by 216°
reflection across line n
none of the above

Explanation:

Step1: Properties of regular hexagon

A regular hexagon has rotational symmetry of order \(6\). The central angle for rotation is \(\frac{360^{\circ}}{n}\), where \(n = 6\) (number of sides). So the angle of rotation for mapping the hexagon onto itself is \(k\times60^{\circ}\), \(k\in\mathbb{Z}\).

• For a \(240^{\circ}\) counter - clockwise rotation (\(240\div60 = 4\)), since \(240 = 4\times60\), it maps the hexagon onto itself.
• For a \(216^{\circ}\) clockwise rotation, \(216\div60=3.6\), not an integer multiple of \(60^{\circ}\), so it does not map the hexagon onto itself.
• A regular hexagon has reflection symmetry. Lines of symmetry for a regular hexagon: lines that pass through opposite vertices and lines that pass through the mid - points of opposite sides. Line \(m\) (passes through mid - points of opposite sides) and line \(n\) (passes through a vertex and mid - point of a side) are lines of symmetry. A reflection across a line of symmetry maps the hexagon onto itself.

Answer:

Reflection across line \(m\), Counterclockwise rotation about \(Y\) by \(240^{\circ}\), Reflection across line \(n\)