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# Reflections

## Transformations that turn a figure into its mirror image by flipping it over a line.

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Reflections

Scott looked at the image below and stated that the image was reflected about the \begin{align*}y\end{align*}-axis. Is he correct? Explain.

### Watch This

First watch this video to learn about reflections.

CK-12 Foundation Chapter10ReflectionsA

Then watch this video to see some examples.

CK-12 Foundation Chapter10ReflectionsB

### Guidance

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A reflection is an example of a transformation that takes a shape (called the preimage) and flips it across a line (called the line of reflection) to create a new shape (called the image).

You can reflect a shape across any line, but the most common reflections are the following:

• reflections across the \begin{align*}x\end{align*}-axis: \begin{align*}y\end{align*} values are multiplied by -1.
• reflections across the \begin{align*}y\end{align*}-axis: \begin{align*}x\end{align*} values are multiplied by -1.
• reflections across the line \begin{align*}y=x\end{align*}: \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values switch places.
• reflections across the line \begin{align*}y = -x\end{align*}. \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values switch places and are multiplied by -1.

#### Example A

Describe the reflection shown in the diagram below.

Solution: The shape is reflected across the y-axis. Let’s examine the points of the shapes.

Points on \begin{align*}WXYZ\end{align*} \begin{align*}W(-7, 5)\end{align*} \begin{align*}X(-1, 5)\end{align*} \begin{align*}Y(-2, 1)\end{align*} \begin{align*}Z(-6, 1)\end{align*}
Points on \begin{align*}W^\prime X^\prime Y^\prime Z^\prime\end{align*} \begin{align*}W^\prime(7, 8)\end{align*} \begin{align*}X^\prime (1, 5)\end{align*} \begin{align*}Y^\prime(2, 1)\end{align*} \begin{align*}Z^\prime(6, 1)\end{align*}

In the table above, all of the \begin{align*}x\end{align*}-coordinates are multiplied by -1. Whenever a shape is reflected across the y-axis, it's \begin{align*}x\end{align*}-coordinates will be multiplied by -1.

#### Example B

Describe the reflection of the purple pentagon in the diagram below.

Solution: The pentagon is reflected across the x-axis. Let’s examine the points of the pentagon.

Points on \begin{align*}DEFGH\end{align*} \begin{align*}D(3.5, 2)\end{align*} \begin{align*}E(5.4, 3)\end{align*} \begin{align*}F(5.5, 6)\end{align*} \begin{align*}G(2.3, 6)\end{align*} \begin{align*}H(1.4, 3.2)\end{align*}
Points on \begin{align*}D^\prime E^\prime F^\prime G^\prime H^\prime\end{align*} \begin{align*}D^\prime(3.5, -2)\end{align*} \begin{align*}E^\prime(5.4, -3)\end{align*} \begin{align*}F^\prime(5.5, -6)\end{align*} \begin{align*}G^\prime(2.3, -6)\end{align*} \begin{align*}H^\prime(1.4, -3.2)\end{align*}

In the table above, all of the \begin{align*}x\end{align*}-coordinates are the same but the \begin{align*}y\end{align*}-coordinates are multiplied by -1. This is what will happen anytime a shape is reflected across the x-axis.

#### Example C

Describe the reflection in the diagram below.

Solution: The shape is reflected across the line \begin{align*}y=x\end{align*}. Let’s examine the points of the preimage and the reflected image.

Points on \begin{align*}GHIJKL\end{align*} \begin{align*}G(-1, 1)\end{align*} \begin{align*}H(-1, 2)\end{align*} \begin{align*}I(-4, 2)\end{align*} \begin{align*}J(-4, 8)\end{align*} \begin{align*}K(-5, 8)\end{align*} \begin{align*}L(-5, 1)\end{align*}
Points on \begin{align*}G^\prime H^\prime I^\prime J^\prime K^\prime L^\prime\end{align*} \begin{align*}G^\prime(1, -1)\end{align*} \begin{align*}H^\prime(2, -1)\end{align*} \begin{align*}I^\prime(2, -4)\end{align*} \begin{align*}J^\prime(8, -4)\end{align*} \begin{align*}K^\prime(8, -5)\end{align*} \begin{align*}L^\prime(1, -5)\end{align*}

Notice that all of the points on the preimage reverse order (or interchange) to form the corresponding points on the reflected image. So for example the point \begin{align*}G\end{align*} on the preimage is at (-1, 1) but the corresponding point \begin{align*}G^\prime\end{align*} on the reflected image is at (1, -1). The \begin{align*}x\end{align*} values and the \begin{align*}y\end{align*} values change places anytime a shape is reflected across the line \begin{align*}y=x\end{align*}.

#### Concept Problem Revisited

Scott looked at the image below and stated that the image was reflected across the \begin{align*}y\end{align*}-axis. Is he correct? Explain.

Scott is correct in that the preimage is reflected about the \begin{align*}y\end{align*}-axis to form the translated image. You can tell this because all points are equidistant from the line of reflection. Let’s examine the points of the trapezoid and see.

Point for \begin{align*}ABCD\end{align*} Point for \begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}
\begin{align*}A(-7, 4)\end{align*} \begin{align*}A^\prime(7, 4)\end{align*}
\begin{align*}B(-3, 4)\end{align*} \begin{align*}B^\prime(3, 4)\end{align*}
\begin{align*}C(-1, 1)\end{align*} \begin{align*}C^\prime(1, 1)\end{align*}
\begin{align*}D(-9, 1)\end{align*} \begin{align*}D^\prime(9, 1)\end{align*}

All of the \begin{align*}y\end{align*}-coordinates for the reflected image are the same as their corresponding points in the preimage. However, the \begin{align*}x\end{align*}-coordinates have been multiplied by -1.

### Vocabulary

Image
In a transformation, the final figure is called the image.
Preimage
In a transformation, the original figure is called the preimage.
Transformation
A transformation is an operation that is performed on a shape that moves or changes it in some way. There are four types of transformations: translations, reflections, dilations and rotations.
Reflection
A reflection is an example of a transformation that flips each point of a shape over the same line.

### Guided Practice

1. Describe the reflection of the pink triangle in the diagram below.

2. Describe the reflection of the purple polygon in the diagram below.

3. Describe the reflection of the blue hexagon in the diagram below.

Answers:

1. Examine the points of the preimage and the reflected image.

Points on \begin{align*}LMO\end{align*} \begin{align*}L(-2, 5)\end{align*} \begin{align*}M(6, 1)\end{align*} \begin{align*}O(-5, 1)\end{align*}
Points on \begin{align*}L^\prime M^\prime O^\prime\end{align*} \begin{align*}L^\prime(-2, -5)\end{align*} \begin{align*}M^\prime(6, -1)\end{align*} \begin{align*}O^\prime(-5, -1)\end{align*}

Notice that all of the \begin{align*}y\end{align*}-coordinates of the preimage (purple triangle) are multiplied by -1 to make the reflected image. The line of reflection is the \begin{align*}x\end{align*}-axis.

2. Examine the points of the preimage and the reflected image.

Points on \begin{align*}AGHI\end{align*} \begin{align*}A(3, 7)\end{align*} \begin{align*}G(3, 4)\end{align*} \begin{align*}H(3, 2)\end{align*} \begin{align*}I(8, 2)\end{align*}
Points on \begin{align*}A^\prime G^\prime H^\prime I^\prime\end{align*} \begin{align*}A^\prime(-3, 7)\end{align*} \begin{align*}G^\prime(-3, 4)\end{align*} \begin{align*}H^\prime(-3, 2)\end{align*} \begin{align*}I^\prime(-8, 2)\end{align*}

Notice that all of the \begin{align*}x\end{align*}-coordinates of the preimage (image 1) is multiplied by -1 to make the reflected image. The line of reflection is the \begin{align*}y\end{align*}-axis.

3. Examine the points of the preimage and the reflected image.

Points on \begin{align*}ABCDEF\end{align*} \begin{align*}A(2, 4)\end{align*} \begin{align*}B(5, 4)\end{align*} \begin{align*}C(6, 2)\end{align*} \begin{align*}D(5, 0)\end{align*} \begin{align*}E(2, 0)\end{align*} \begin{align*}F(1, 2)\end{align*}
Points on \begin{align*}A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime\end{align*} \begin{align*}A^\prime(-4, -2)\end{align*} \begin{align*}B^\prime(-4, -5)\end{align*} \begin{align*}C^\prime(-2, -6)\end{align*} \begin{align*}D^\prime(0, -5)\end{align*} \begin{align*}E^\prime(0, -2)\end{align*} \begin{align*}F^\prime(-2, -1)\end{align*}

Notice that both the \begin{align*}x\end{align*}-coordinates and the \begin{align*}y\end{align*}-coordinates of the preimage (image 1) change places to form the reflected image. As well the points are multiplied by -1. The line of reflection is the line \begin{align*}y = -x\end{align*}.

### Practice

If the following points were reflected across the \begin{align*}x\end{align*}-axis, what would be the coordinates of the reflected points? Show these reflections on a graph.

1. (3, 1)
2. (4, -2)
3. (-5, 3)
4. (-6, 4)

If the following points were reflected across the \begin{align*}y\end{align*}-axis, what would be the coordinates of the reflected points? Show these reflections on a graph.

1. (-4, 3)
2. (5, -4)
3. (-5, -4)
4. (3, 3)

If the following points were reflected about the line \begin{align*}y=x\end{align*}, what would be the coordinates of the reflected points? Show these reflections on a graph.

1. (3, 1)
2. (4, -2)
3. (-5, 3)
4. (-6, 4)

Describe the following reflections:

### Vocabulary Language: English

Coordinate Plane

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

Geometric patterns are visual patterns of geometric figures that follow a rule.
Image

Image

The image is the final appearance of a figure after a transformation operation.
perpendicular bisector

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

Perpendicular lines are lines that intersect at a $90^{\circ}$ angle.
Preimage

Preimage

The pre-image is the original appearance of a figure in a transformation operation.
Reflection

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

Transformation

A transformation moves a figure in some way on the coordinate plane.
Rigid Transformation

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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