Refer to the figure below, which shows the reflection of about ; we will call this image .

Note that coincides with . Now, refer to the figure below, which shows the reflection of about ; we will call this image - the final image -

Note that coincides with , making this the correct response.

Reflecting a point

over the y-axis geometrically is the same as negating the x-coordinate of the ordered pair to obtain

.

Thus, since our initial point was

and we want to reflect it over the y-axis, we obtain the reflection by negating the first term of the ordered pair to get

.

Refer to the figure below, which shows the reflection of the given hexagon about ; we will call the image of , call the image of , and so forth.

Now, refer to the figure below, which shows the reflection of the image about ; we will call the image of , call the image of , and so forth.

Note that the vertices coincide with those of the original hexagon, and that the images of the points are in the same clockwise order as the original points. Since coincides with , coincides with , and so forth, a clockwise rotation of five-sixth of a complete turn - that is, ,

is required to make each point its own image under the three transformations.

A line of symmetry of a figure is about which the reflection of the figure is the figure itself. The diagram below shows the only line of symmetry of the figure.

The correct response is one.

The reflection of the point at about the -axis is the point at ; therefore, the image of is . Since these two points have the same -coordinate, the distance between them is the absolute value of the difference between their -coordinates: