Question

what set of reflections would carry hexagon abcdef onto itself? image of hexagon with coordinates and axes options:

• x-axis, ( y = x ), x-axis, ( y = x )
• y-axis, x-axis, y-axis
• x-axis, y-axis, y-axis
• ( y = x ), x-axis, ( y = x ), y-axis

Explanation:

Step1: Analyze the hexagon's symmetry

The hexagon is symmetric about the y - axis first. Reflecting over the y - axis will map the hexagon to a position, then reflecting over the x - axis will further transform it, and then reflecting over the y - axis again will bring it back to its original position. Let's check each option:

• Option 1: Repeating \(x -\)axis and \(y = x\) reflections won't map the hexagon to itself as the hexagon is not symmetric about \(y=x\) in a way that these repeated reflections work.
• Option 2: Reflect over \(y -\)axis (maps the hexagon to a symmetric position), then \(x -\)axis (transforms it), then \(y -\)axis (brings it back to original). Let's verify with a vertex. Take point \(A(0,1)\). Reflect over \(y -\)axis: \(A\) becomes \((0,1)\) (wait, no, if the hexagon is centered? Wait, looking at the coordinates, the hexagon is symmetric about the y - axis (the vertical line \(x = - 0.5\)? Wait, maybe the y - axis here is the vertical line of symmetry. Let's take a general approach. The correct sequence for a symmetric figure (like this hexagon which is symmetric about y - axis, then x - axis related) is \(y -\)axis, \(x -\)axis, \(y -\)axis.
• Option 3: \(x -\)axis, \(y -\)axis, \(y -\)axis won't work as the first \(x -\)axis reflection will not be undone properly.
• Option 4: Reflections over \(y=x\) and \(x -\)axis and \(y=x\) and \(y -\)axis are not the correct symmetries for this hexagon.

Step2: Confirm the correct option

By analyzing the symmetry of the hexagon (it is symmetric about the y - axis, then x - axis, then y - axis to map onto itself), the set of reflections \(y -\)axis, \(x -\)axis, \(y -\)axis is correct.

Answer:

\(y\)-axis, \(x\)-axis, \(y\)-axis