# 12.5: Reflections

Difficulty Level: At Grade Created by: CK-12
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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.) After completing this Concept, you'll be able to answer this question.

### Watch This

Watch the last part of this video.

### Guidance

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 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 vertical 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 line , then the image is .
• Reflection over : If is reflected over the line , then the image is .

#### Example A

Reflect the letter 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.

#### Example B

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.

#### Example C

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.

#### Example D

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.

Watch this video for help with the Examples above.

#### Concept 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.

### Vocabulary

A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation) is a transformation that does not change the size or shape of a figure. The new figure created by a transformation is called the image. The original figure is called the preimage. A reflection is a transformation that turns a figure into its mirror image by flipping it over a line. The line of reflection is the line that a figure is reflected over.

### Guided Practice

1. Reflect the line segment with endpoints and over the line .

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

3. Reflect the trapezoid TRAP over the line .

1. 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.

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

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

### Practice

1. Which letter is a reflection over a vertical line of the letter ?
2. Which letter is a reflection over a horizontal line of the letter ?

Reflect each shape over the given line.

1. axis
2. 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.

1. Plot on the coordinate plane.
2. Reflect over . Find the coordinates of .
3. Reflect over . Find the coordinates of .
4. What one transformation would be the same as this double reflection?

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

1. Plot on the coordinate plane.
2. Reflect over . Find the coordinates of .
3. Reflect over . Find the coordinates of .
4. What one transformation would be the same as this double reflection?

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

1. Plot on the coordinate plane.
2. Reflect over the axis. Find the coordinates of .
3. Reflect over the axis. Find the coordinates of .
4. What one transformation would be the same as this double reflection?

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Coordinate Plane The coordinate plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin. The coordinate plane is also called a Cartesian Plane.
Geometric Patterns Geometric patterns are visual patterns of geometric figures that follow a rule.
Image The image is the final appearance of a figure after a transformation operation.
perpendicular bisector A perpendicular bisector of a line segment passes through the midpoint of the line segment and intersects the line segment at $90^\circ$.
Perpendicular lines Perpendicular lines are lines that intersect at a $90^{\circ}$ angle.
Preimage The pre-image is the original appearance of a figure in a transformation operation.
Reflection A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.
Transformation A transformation moves a figure in some way on the coordinate plane.
Rigid Transformation A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

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