Examine the figure below:

If we connect the horizontal line with the line along the rotated nine at right, we see that it is the result of a one-third turn clockwise; the angle between them

Examine the figure below.

If we connect the horizontal line with the line along the rotated "omega" at right, we see that it is the result of a one-sixth turn counterclockwise; the angle between them is one sixth of , or .

To perform a dilation with center and scale factor , find the midpoints of the segments connecting to each point, and connect those points. We can simplify the process by finding the midpoints of , , and , and naming them , , and , respectively; , the image of center , is just itself. The figure is below:

Now, do the same thing to the new square, but with as the center. The figure is below:

The final image, relative to the original square, is below:

A dilation is the stretching or shrinking of a figure.

A rotation is the turning of a a figure about a point.

A reflection is the flipping of a figure across a line.

A translation is is the sliding of a figure in a direction.

With a translation, the image is not only congruent to its original size and shape, but its orientation remains the same. A translation fits this figure best because the shape seems to move upward and rightward without changing size, shape, or orientation.

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.

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

Examine the figure below.

If we connect the horizontal line with the line along the rotated "omega" at right, we see that it is the result of a one-sixth turn counterclockwise; the angle between them is one sixth of , or .

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 .