What if you noticed that a lake can act like a mirror in nature? Describe the line of reflection in the photo below. If this image were on the coordinate plane, what could the equation of the line of reflection be? (There could be more than one correct answer, depending on where you place the origin.)

### Reflections

A **transformation** is an operation that moves, flips, or changes a figure to create a new figure. A **rigid transformation** is a transformation that preserves size and shape. The rigid transformations are: translations, reflections (discussed here), and rotations. The new figure created by a transformation is called the **image**. The original figure is called the **preimage**. Another word for a rigid transformation is an **isometry**. Rigid transformations are also called **congruence transformations**. If the preimage is , then the image would be labeled , said “a prime.” If there is an image of , that would be labeled , said “a double prime.”

A **reflection** is a transformation that turns a figure into its mirror image by flipping it over a line. Another way to describe a reflection is a “flip.” The **line of reflection** is the line that a figure is reflected over. If a point is on the line of reflection then the image is the same as the original point.

##### Common Reflections

**Reflection over :**If is reflected over the line , then the image is .**Reflection over**If is reflected over the horizontal line , then the image is .**Reflection over**If is reflected over the vertical line , then the image is .**Reflection over the axis:**If is reflected over the axis, then the image is .**Reflection over the axis:**If is reflected over the axis, then the image is .**Reflection over :**If is reflected over the line , then the image is .

#### Performing a Reflection

Reflect the letter ‘‘F’’ over the axis.

To reflect the letter over the axis, now the coordinates will remain the same and the coordinates will be the same distance away from the axis on the other side.

#### Reflecting a Triangle

1. Reflect over the axis. Find the coordinates of the image.

To reflect over the axis the coordinates will remain the same. The coordinates will be the same distance away from the axis, but on the other side of the axis.

2. Reflect the triangle with vertices and over the line .

Notice that this vertical line is through our preimage. Therefore, the image’s vertices are the same distance away from as the preimage. As with reflecting over the axis (or ), the coordinates will stay the same.

#### Reflecting a Square

Reflect square over the line .

The purple line is . To reflect an image over a line that is not vertical or horizontal, you can fold the graph on the line of reflection.

#### Lake Problem Revisited

The white line in the picture is the line of reflection. This line coincides with the water’s edge. If we were to place this picture on the coordinate plane, the line of reflection would be any horizontal line. One example could be the axis.

### Examples

#### Example 1

Reflect the line segment with endpoints and over the line .

Here, the line of reflection is on , which means has the same coordinates. has the same coordinate as and is the same distance away from , but on the other side.

#### Example 2

A triangle and its reflection, are to the left. What is the line of reflection?

Looking at the graph, we see that the preimage and image intersect when . Therefore, this is the line of reflection.

#### Example 3

Reflect the trapezoid *TRAP* over the line .

The purple line is . You can reflect the trapezoid over this line just like we did in Example D.

### Review

- Which letter is a reflection over a vertical line of the letter ?
- Which letter is a reflection over a horizontal line of the letter ?

Reflect each shape over the given line.

- axis
- axis

Find the line of reflection of the blue triangle (preimage) and the red triangle (image).

** Two Reflections** The vertices of are , and . Use this information to answer questions 10-13.

- Plot on the coordinate plane.
- Reflect over . Find the coordinates of .
- Reflect over . Find the coordinates of .
- What
transformation would be the same as this double reflection?*one*

** Two Reflections** The vertices of are , and . Use this information to answer questions 14-17.

- Plot on the coordinate plane.
- Reflect over . Find the coordinates of .
- Reflect over . Find the coordinates of .
- What
transformation would be the same as this double reflection?*one*

** Two Reflections** The vertices of are , and . Use this information to answer questions 18-21.

- Plot on the coordinate plane.
- Reflect over the axis. Find the coordinates of .
- Reflect over the axis. Find the coordinates of .
- What
transformation would be the same as this double reflection?*one*

### Review (Answers)

To view the Review answers, open this PDF file and look for section 12.5.