The midpoint of a segment with endpoints at  and is located at

Substitute the coordinates of  and  in this formula to find that midpoint of is located at

, or .

Substitute the coordinates of and to find that midpoint of is located at

, or .

To perform the translation , or, equivalently,

,

on a point, it is necessary to add

to its -coordinate, and

to its -coordinate.

Therefore, the -coordinate of the image of  under this translation is

;

its -coordinate is

The image of is located at .

The translation  on a figure is the translation that shifts a figure so that the image of , which we will call , coincides with . All other points shift the same distance in the same direction. Below shows the image of the given hexagon under this translation, with the image of  marked as :

If the image is reflected about , the new image is the original hexagon. Calling  the image of  under this reflection, we get the following:

, the image of  under these two transformations, coincides with .

The translation  on a figure is the translation that shifts a figure so that the image of , which we will call , coincides with . All other points shift the same distance in the same direction. Below shows the image of the given hexagon under this translation:

If this new hexagon is rotated clockwise  - one third of a turn - about , and call  the image of , and so forth, the result is as follows:

Removing the intermediate markings, we see that the correct response is 

The translation  on a figure is the translation that shifts a figure so that the image of  coincides with . All other points shift the same distance in the same direction. Below shows the image of the given hexagon under this translation, with  the image of :

If this new hexagon is rotated  - one half of a turn - about  - the image is the original hexagon, but the vertices can be relabeled. Letting  be the image of  under this rotation, and so forth:

 coincides with  in the original hexagon, making  the correct response.

By definition, a median of a triangle has as its endpoints one vertex and the midpoint of the opposite side. Therefore, the endpoints of the median from are itself, which is at , and , which itself is the midpoint of the side with origin  and , which is , as its endpoints.

The midpoint of a segment with endpoints at  and is located at

,

so, substituting the coordinates of and in the formula, we see that is

, or .

See the figure below:

To perform the translation , or, equivalently,

,

on a point, it is necessary to add

and

to the - and - coordinates, respectively. Therefore, the image of  is located at 

,

or

.

The translation  on a figure is the translation that shifts a figure so that the image of  coincides with . All other points shift the same distance in the same direction. Below shows the image of the given hexagon under this translation, with the image of  marked as :

If this new hexagon is rotated clockwise  - one third of a turn - about  - the image is the original hexagon, but the vertices can be relabeled. Letting  be the image of  under this rotation, and so forth:

 coincides with  in the original hexagon, making  the correct response.

Rotating a point 

geometrically in the plane about the origin  is equivalent to negating the coordinates of the point algebraically to obtain

.

Thus, since our initial point was

we negate both coordinates to get

as the rotation about the origin by .

The diagram below superimposes the two figures:

The transformation moves the black diagonal to the position of the red diagonal, and, consequently, points  and  to points  and , respectively. This constitutes two-tenths of a complete turn clockwise, or a clockwise rotation of 

A counterclockwise rotation of  is  ofa complete rotation. Observe the following diagram:

In the right figure, the question mark has been turned one-eighth of a complete turn counterclockwise. This is the correct orientation.

A  rotation is equivalent to  of a complete rotation, so rotate as follows:

The image of  under this rotation, which we will call , is at .

Now, locate the midpoints  and , and construct the line  as described and shown below. Perform the reflection:

It can be seen that the image of  under this transformation - the desired  - is located at .