# HiSET: Math : Reflections

## Example Questions

### Example Question #1 : Reflections

What is the result of reflecting the point  over the y-axis in the coordinate plane?

Explanation:

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

.

### Example Question #2 : Reflections

How many lines of symmetry does the above figure have?

Infinitely many

None

Four

One

Two

One

Explanation:

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.

### Example Question #3 : Reflections

On the coordinate plane, the point  is reflected about the -axis. The image is denoted . Give the distance .

Explanation:

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:

### Example Question #4 : Reflections

Consider regular Hexagon ; let and be the midpoints of and . Reflect the hexagon about , then again about . Which of the following clockwise rotations about the center would result in each point being its own image under this series of transformations?

Explanation:

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.

### Example Question #5 : Reflections

Consider regular Hexagon ; let and be the midpoints of and . Reflect the hexagon about , then again about . With which of the following points does the image of under these reflections coincide?

Explanation:

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.