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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. 

Reflection

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.

  

 

Let's describe the reflections shown below:

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.

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.

    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*}.

    Examples

    Example 1

    Earlier, you were told that 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.

    Example 2

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

    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.

    Example 3

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

    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.

    Example 4

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

    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*}.

    Review

    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:

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 10.4. 

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    Vocabulary

    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.

    Image Attributions

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