10.2: Reflections of Geometric Shapes
Objectives
The lesson objectives for Reflections of Geometric Shapes are:
- What is a reflection?
- Graphing an image that has undergone a reflection
- Writing a rule for a reflection
- The properties of a reflection
What Is a Reflection?
Concept Content
As you learned in the previous lesson, there are four types of transformations. In the previous lesson, you learned about one of these transformations, translations. The other three transformations are reflections, dilations and rotations. In this lesson, you will learn about reflections. Reflections in transformations involve flipping a shape or figure. Reflections can be performed over a line of reflection, a point of reflection, or a plane of reflection. In reflections, all of the points in the preimage and the reflected image are equidistant from this line of reflection. As well, since reflections flip over the line of reflection (or the mirror line) the points on the mirror line stay the same for a reflection and the other points are on the opposite side.
In this concept you will learn to describe reflections and, in the process, learn what they are. You will concentrate on four types of reflections:
- reflections in the \begin{align*}x\end{align*}-axis,
- reflections in the \begin{align*}y\end{align*}-axis,
- reflections about the origin, and
- reflections in the line \begin{align*}y = x\end{align*} or \begin{align*}y = -x\end{align*}.
In later concepts, you will also learn to graph reflections, write rules to describe them, and finally learn some of the properties of reflections.
Guidance
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.
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*} |
If you notice in the table, 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*}-coordinate has been multiplied by –1. The mirror line (or the line of reflection) is the \begin{align*}y\end{align*}-axis. Scott knows that in reflections the points on the mirror line stay the same and the other points are on the opposite side. Therefore the \begin{align*}y\end{align*}-coordinates would remain the same (since the \begin{align*}y\end{align*}-axis is the mirror line) and the \begin{align*}x\end{align*}-coordinates would change sign.
Examples
Example A
Describe the reflection of the light blue triangle in the diagram below.
Let’s examine the points in the triangle.
Points on \begin{align*}BCD\end{align*} | \begin{align*}B(-4, 2)\end{align*} | \begin{align*}C(-2, -2)\end{align*} | \begin{align*}D(-7, -2)\end{align*} |
Points on \begin{align*}B^\prime C^\prime D^\prime\end{align*} | \begin{align*}B^\prime(4, -2)\end{align*} | \begin{align*}C^\prime (2, 2)\end{align*} | \begin{align*}D^\prime(7, 2)\end{align*} |
In the table above, all of the \begin{align*}x\end{align*}-coordinates and the \begin{align*}y\end{align*}-coordinates are multiplied by –1. This would tell you that the line of reflection (or the mirror line) is the origin point (0, 0).
Example B
Describe the reflection of the purple pentagon in the diagram below.
Let’s examine the points in 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 would tell you that the line of reflection (or the mirror line) is the \begin{align*}x\end{align*}-axis.
Example C
Describe the reflection in the diagram below.
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*} |
The line of reflection, or the mirror line is \begin{align*}y = x\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). Therefore the \begin{align*}x\end{align*} values and the \begin{align*}y\end{align*} values change places.
Vocabulary
- Image
- In transformations, the final figure is called the image.
- Mirror Line
- A mirror line is another term for the line of reflection.
- Preimage
- In transformations, the original image \begin{align*}(ABCD)\end{align*} is called a preimage.
- Transformation
- A transformation is an operation that is created from some original image. There are four types of transformations: translations, reflections, dilations and rotations.
- Reflections
- Reflections in transformations involve flipping a shape or figure over a line of reflection, a point of reflection, or a plane of reflection. In reflections, all of the points in the preimage and the translated image are equidistant from this line of reflection.
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. Therefore, the line of reflection, or the mirror line, 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. Therefore, the line of reflection, or the mirror line, 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. Therefore, the line of reflection, or the mirror line, is the line \begin{align*}y = -x\end{align*}.
Summary
In this concept you began your study of reflections, the second type of transformation. Remember reflections involve flipping an image over a line, a point, or a plane. The reflection line is often called the mirror line. You learned the following types of reflections in this concept:
Type of Reflection | What did you learn... |
---|---|
Reflections in the \begin{align*}x\end{align*}-axis | \begin{align*}x\end{align*} values stay the same, \begin{align*}y\end{align*} values multiply by –1 |
Reflections in the \begin{align*}y\end{align*}-axis | \begin{align*}y\end{align*} values stay the same, \begin{align*}x\end{align*} values multiply by –1 |
Reflections about the origin | Both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values multiply by –1 |
Reflections in the line \begin{align*}y = x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image |
Reflections in the line \begin{align*}y = -x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image and multiply them by –1 |
Being able to recognize a reflection (or a flip) allows you to describe this type of transformation of the preimage to the transformed image. In the next concept you will learn how to graph reflection images.
Problem Set
If the following points were reflected about the \begin{align*}x\end{align*}-axis, what would be the coordinates of the reflected points? Show these reflections on a graph.
- (3, 1)
- (4, –2)
- (–5, 3)
- (–6, 4)
If the following points were reflected about the \begin{align*}y\end{align*}-axis, what would be the coordinates of the reflected points? Show these reflections on a graph.
- (–4, 3)
- (5, –4)
- (–5, –4)
- (3, 3)
If the following points were reflected about the origin, what would be the coordinates of the reflected points? Show these reflections on a graph.
- (3, –1)
- (–6, 2)
- (–7, 3)
- (–8, –9)
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.
- (3, 1)
- (4, –2)
- (–5, 3)
- (–6, 4)
Describe the following reflections:
Graphing an Image that has undergone a Reflection
Concept Content
In this second concept of lesson Reflections of Geometric Shapes, you will learn how to graph an image that has undergone a reflection. Remember that a reflection takes place when a preimage is flipped over a line, a point, or a plane in the Cartesian plane to form a new image.
In the last lesson you learned to recognize what happens to the points and images when reflected in different lines or points. In this lesson you will create the images and preimages by first graphing them on a Cartesian plane. When graphing the reflected image, it is often helpful to remember that what happens to the points in the preimage depends on the type of reflection. The table below summarizes the types of reflections and what happens to the points in your preimage based on each type.
Type of Reflection | What did you learn... |
---|---|
Reflections in the \begin{align*}x\end{align*}-axis | \begin{align*}x\end{align*} values stay the same, \begin{align*}y\end{align*} values multiply by –1 |
Reflections in the \begin{align*}y\end{align*}-axis | \begin{align*}y\end{align*} values stay the same, \begin{align*}x\end{align*} values multiply by –1 |
Reflections about the origin | Both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values multiply by –1 |
Reflections in the line \begin{align*}y = x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image |
Reflections in the line \begin{align*}y = -x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image and multiply them by –1 |
Guidance
Triangle \begin{align*}A\end{align*} has coordinates \begin{align*}E(-5, -5)\end{align*}, \begin{align*}F(2, -6)\end{align*} and \begin{align*}G(-2, 0)\end{align*}. Draw the triangle on the Cartesian plane. Reflect the image in the \begin{align*}y\end{align*}-axis. State the coordinates of the resulting image.
The coordinates of the new image \begin{align*}(B)\end{align*} are \begin{align*}E^\prime(5, -5)\end{align*}, \begin{align*}F^\prime(-2, -6)\end{align*} and \begin{align*}G^\prime(2, 0)\end{align*}.
Examples
Example A
Line \begin{align*}\overline{AB}\end{align*} drawn from (–4, 2) to (3, 2) has been reflected in the \begin{align*}x\end{align*}-axis. Draw the preimage and image and properly label each.
Example B
The diamond \begin{align*}ABCD\end{align*} is reflected about the line \begin{align*}y = x\end{align*} to form the image \begin{align*}A^\prime B^\prime C^\prime D^\prime \end{align*}. Find the coordinates of the reflected image. On the diagram, draw and label the reflected image.
Example C
The following figure is reflected about the origin to make a translated image. Find the coordinates of the reflected image. On the diagram, draw and label the image.
Vocabulary
- Image
- In transformations, the final figure is called the image.
- Mirror Line
- A mirror line is another term for the line of reflection.
- Preimage
- In transformations, the original image \begin{align*}(ABCD)\end{align*} is called a preimage.
- Transformation
- A transformation is an operation that is created from some original image. There are four types of transformations: translations, reflections, dilations and rotations
- Reflections
- Reflections in transformations involve flipping a shape or figure over a line of reflection, a point of reflection, or a plane of reflection. In reflections, all of the points in the preimage and the translated image are equidistant from this line of reflection.
Guided Practice
1. Line \begin{align*}\overline{ST}\end{align*} is drawn from (–3, 4) to (–3, 8) has been reflected about the line \begin{align*}y = -x\end{align*}. Draw the preimage and image and properly label each.
2. The polygon below has been reflected in the \begin{align*}y\end{align*}-axis. Draw the reflected image and properly label each.
3. The purple pentagon is reflected in the \begin{align*}y-axis\end{align*} to make the new image. Find the coordinates of the purple pentagon. On the diagram, draw and label the reflected pentagon.
Answers
1.
2.
3.
Summary
In this second concept of Chapter Is it a Slide, a Flip, or a Turn? you were introduced to describing reflections, keeping in mind the way coordinates change when reflecting about a line or a point. In this lesson you worked from the reflection description and graphed the reflected image. You used the following concepts to graph the reflected images:
Line of reflection | Points on preimage | Points on reflected image |
---|---|---|
\begin{align*}x\end{align*}-axis | \begin{align*}(x, y)\end{align*} | \begin{align*}(x, -y)\end{align*} |
\begin{align*}y\end{align*}-axis | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, y)\end{align*} |
\begin{align*}y = x\end{align*} | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, x)\end{align*} |
\begin{align*}y = -x\end{align*} | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, -x)\end{align*} |
\begin{align*}(0, 0)\end{align*} | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} |
Problem Set
- Reflect figure \begin{align*}MATH\end{align*} on the grid about the \begin{align*}y\end{align*}-axis.
- Reflect figure \begin{align*}DEFGH\end{align*} about the line \begin{align*}y=x\end{align*}.
- Reflect figure \begin{align*}LMO\end{align*} in the \begin{align*}x\end{align*}-axis.
- Reflect figure \begin{align*}WXY\end{align*} about the origin.
- Reflect the purple figure in the line \begin{align*}y=-x\end{align*}.
For each of the following diagrams reflect the images about the \begin{align*}y\end{align*}-axis. Find the coordinates of the preimages. On each diagram, draw and label the reflected image.
For each of the following diagrams reflect the images about the line \begin{align*}y=x\end{align*}. Find the coordinates of the preimages. On each diagram, draw and label the reflected image.
Writing a Rule for a Reflection
Concept Content
In this third concept with respect to reflections, you will learn how to describe a reflection using a rule. In the first lesson you used words to describe a reflection. By examining the coordinates of the reflected image, you could describe if the preimage was reflected in the \begin{align*}x\end{align*}-axis, the \begin{align*}y\end{align*}-axis, the origin, or the lines \begin{align*}y=x\end{align*} or \begin{align*}y=-x\end{align*}.
You could have used a notation to describe the reflection. If, for instance you had the image below:
You would notice that the preimage is reflected in the \begin{align*}y\end{align*}-axis. If you were to describe the reflected image using notation, you would write the following:
\begin{align*}P(x,y) \rightarrow P^\prime (-x,y)\end{align*} or \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}
In this concept you will write rules for reflection as well as draw graphs from such rules. For this concept you will be using the second of the above notations, namely \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}.
Guidance
The figure below shows a pattern of two fish. Write the mapping rule for the reflection of Image A to Image B.
To answer this question, look at the coordinate points for Image A and Image B.
Image A | \begin{align*}A(-11.8, 5)\end{align*} | \begin{align*}B(-11.8, 2)\end{align*} | \begin{align*}C(-7.8, 5)\end{align*} | \begin{align*}D(-4.9, 2)\end{align*} | \begin{align*}E(-8.7, 0.5)\end{align*} | \begin{align*}F(-10.4, 3.1)\end{align*} |
Image B | \begin{align*}A^\prime (-11.8,-5)\end{align*} | \begin{align*}B^\prime(-11.8, -2)\end{align*} | \begin{align*}C^\prime(-7.8, -5)\end{align*} | \begin{align*}D^\prime(-4.9, -2)\end{align*} | \begin{align*}E^\prime(-8.7, -0.5)\end{align*} | \begin{align*}F^\prime(-10.4, -3.1)\end{align*} |
Notice that all of the \begin{align*}y\end{align*}-coordinates have changed sign. Therefore Image A has reflected across the \begin{align*}x\end{align*}-axis. To write a rule for this reflection you would write: \begin{align*}r_{x-axis}(x,y) \rightarrow (x,-y)\end{align*}.
Examples
Example A
Find an image of the point (3, 2) that has undergone a reflection in:
a) the \begin{align*}x\end{align*}-axis,
b) the \begin{align*}y\end{align*}-axis,
c) the line \begin{align*}y=x\end{align*}, and
d) the line \begin{align*}y=-x\end{align*}.
Write the notation to describe the reflection.
a) Reflection in the \begin{align*}x\end{align*}-axis: \begin{align*}r_{x-axis}(3,2) \rightarrow (3,-2)\end{align*}
b) Reflection in the \begin{align*}y\end{align*}-axis: \begin{align*}r_{y-axis}(3,2) \rightarrow (-3,2)\end{align*}
c) Reflection in the \begin{align*}y=x\end{align*}: \begin{align*}r_{y=x}(3,2) \rightarrow (2,3)\end{align*}
d) Reflection in the \begin{align*}y=x\end{align*}: \begin{align*}r_{y=-x}(3,2) \rightarrow (-2,-3)\end{align*}
Example B
Reflect Image A in the diagram below:
a) About the \begin{align*}y\end{align*}-axis and label it \begin{align*}B\end{align*}.
b) About the \begin{align*}x\end{align*}-axis and label it \begin{align*}O\end{align*}.
c) In the line \begin{align*}y=-x\end{align*} and label it \begin{align*}Z\end{align*}.
Write notation for each to indicate the type of reflection.
a) Reflection in the \begin{align*}y\end{align*}-axis: \begin{align*}r_{y-axis}A \rightarrow B=r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}
b) Reflection in the \begin{align*}x\end{align*}-axis: \begin{align*}r_{x-axis}A \rightarrow O=r_{x-axis}(x,y) \rightarrow (x,-y)\end{align*}
c) Reflection in the \begin{align*}y=-x\end{align*}: \begin{align*}r_{y=-x}A \rightarrow Z=r_{y=-x}(x,y) \rightarrow (-y,-x)\end{align*}
Example C
Write the notation that represents the reflection of the preimage A to the reflected image J in the diagram below.
First, pick a point in the diagram to use to see how it is reflected.
\begin{align*}D: (-1, 5) \quad D^\prime: (1, -5)\end{align*}
Notice how both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} is multiplied by –1. This indicates that the preimage A is reflected about the origin to form the Reflected Image J. therefore the notation is \begin{align*}r_{origin}A \rightarrow J=r_{origin}(x,y) \rightarrow (-x,-y)\end{align*}.
Vocabulary
- Notation Rule
- A notation rule has the following form \begin{align*}r_{y-axis}A \rightarrow B=r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*} and tells you that the image A has been reflected in the \begin{align*}y\end{align*}-axis and only the \begin{align*}x\end{align*}-coordinates are multiplied by –1. The \begin{align*}y\end{align*}-coordinates remain the same.
- Reflections
- Reflections in transformations involve flipping a shape or figure over a line of reflection, a point of reflection, or a plane of reflection. In reflections, all of the points in the preimage and the translated image are equidistant from this line of reflection.
Guided Practice
1. Thomas describes a translation as point \begin{align*}J\end{align*} moving from \begin{align*}J(-2, 6)\end{align*} to \begin{align*}J^\prime(-2, -6)\end{align*}. Write the notation to describe this reflection for Thomas.
2. Write the notation that represents the reflection of the yellow diamond to the reflected green diamond in the diagram below.
3. Karen was playing around with a drawing program on her computer. She created the following diagrams and ten wanted to determine the transformations. Write the notation rule that represents the transformation of the purple and blue diagram to the orange and blue diagram.
Answers
1. \begin{align*}J: (-2, 6) \quad J^\prime: (-2, -6)\end{align*}
Since the \begin{align*}y\end{align*}-coordinate is multiplied by –1 and the \begin{align*}x\end{align*}-coordinate remains the same, this is a reflection in the \begin{align*}x\end{align*}-axis. The notation is: \begin{align*}r_{x-axis}J \rightarrow J^\prime=r_{x-axis} (-2,6) \rightarrow (-2,-6)\end{align*}
2. In order to write the notation to describe the reflection, choose one point on the preimage (the yellow diamond) and then the reflected point on the green diamond to see how the point has moved. Notice that point E is shown in the diagram:
\begin{align*}E(-1,3) \rightarrow E^\prime(3,-1)\end{align*}
Since both \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates are reversed numbers, the reflection is in the line \begin{align*}y=x\end{align*}. The notation for this reflection would be: \begin{align*}r_{y=x}(x,y) \rightarrow (y,x)\end{align*}.
3. In order to write the notation to describe the transformation, choose one point on the preimage (purple and blue diagram) and then the transformed point on the orange and blue diagram to see how the point has moved. Notice that point \begin{align*}A\end{align*} is shown in the diagram:
\begin{align*}C(7,0) \rightarrow C^\prime(-7,0)\end{align*}
Since both \begin{align*}x\end{align*}-coordinates only are multiplied by –1, the transformation is a reflection is in \begin{align*}y\end{align*}-axis. The notation for this reflection would be: \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*}.
Summary
In this concept of lesson Reflections of Geometric Shapes you have learned to describe a reflection in notation form. Notations for reflections can allow you to quickly determine where the reflected point or points are going to be in a diagram. In other words, you can draw a reflected image from a preimage when you know the notation. In order to write a notation you use the following general forms:
Line of reflection | Points on preimage | Points on reflected image | Notation rule |
---|---|---|---|
\begin{align*}x\end{align*}-axis | \begin{align*}(x, y)\end{align*} | \begin{align*}(x, -y)\end{align*} | \begin{align*}r_{x-axis}(x,y) \rightarrow (x,-y)\end{align*} |
\begin{align*}y\end{align*}-axis | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, y)\end{align*} | \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*} |
\begin{align*}y = x\end{align*} | \begin{align*}(x, y)\end{align*} | \begin{align*}(y, x)\end{align*} | \begin{align*}r_{y=x}(x,y) \rightarrow (y,x)\end{align*} |
\begin{align*}y = -x\end{align*} | \begin{align*}(x, y)\end{align*} | \begin{align*}(-y, -x)\end{align*} | \begin{align*}r_{y=-x}(x,y) \rightarrow (-y,-x)\end{align*} |
\begin{align*}(0, 0)\end{align*} | \begin{align*}(x, y)\end{align*} | \begin{align*}(-x, -y)\end{align*} | \begin{align*}r_{origin}(x,y) \rightarrow (-x,-y)\end{align*} |
Using these mapping rules allows you to quickly determine the types of reflections in images. It also allows you to draw a reflected image from the notation.
Problem Set
Write the notation to describe the movement of the points in each of the reflections below.
- \begin{align*}S(1,5) \rightarrow S^\prime(-1,5)\end{align*}
- \begin{align*}W(-5,-1) \rightarrow W^\prime(5,1)\end{align*}
- \begin{align*}Q(2,-5) \rightarrow Q^\prime(2,5)\end{align*}
- \begin{align*}M(4,3) \rightarrow M^\prime(-3,-4)\end{align*}
- \begin{align*}B(-4,-2) \rightarrow B^\prime(-2,-4)\end{align*}
Write the notation that represents the reflection of the preimage A to the reflected images in the diagrams below.
The Properties of a Reflection
Concept Content
As you learned earlier, reflections involve flips of a preimage to form a reflected image. In this last concept you will look at the properties of reflections. Reflections result in reverse orientation. As you have seen in the examples from the previous three concepts, a reflection involves flipping an image or a point in a line or over the origin. The line is known as the line of symmetry. The line of symmetry is really just a line of reflection but it allows the point's preimage to be reflected to a translated image and maintain its measurements. Therefore reflected images have the same length and angle measurements as their preimages. As well, points of the reflected image are collinear if they are collinear in the preimage. In other words, if \begin{align*}B\end{align*} is between \begin{align*}A\end{align*} and \begin{align*}C\end{align*}, then \begin{align*}B^\prime\end{align*} will be between \begin{align*}A^\prime\end{align*} and \begin{align*}C^\prime\end{align*}.
In summary, reflections are considered to have the following properties:
- line segments are the same length (distance)
- angle measures in a figure remain the same (angles)
- reverse orientation
- points that are on each line remain on the line for the reflected image (collinear)
Guidance
The triangle \begin{align*}ABC\end{align*} is drawn such that the vertices are at \begin{align*}A(1, 1)\end{align*}, \begin{align*}B(5, 5)\end{align*} and \begin{align*}C(-2, 4)\end{align*}. Triangle \begin{align*}TOM\end{align*} is reflected across the \begin{align*}x\end{align*}-axis.
a) What are the coordinates of Triangle \begin{align*}TOM\end{align*}?
b) Measure each angle either with a protractor or using geometry software. What angles are congruent?
c) Measure each side length. Are the distances the same?
d) Can you conclude that the triangle \begin{align*}TOM\end{align*} is a reflection of triangle \begin{align*}ABC\end{align*}?
a) The coordinates of \begin{align*}TOM\end{align*} upon reflection in the \begin{align*}x\end{align*}-axis would be:
Preimage \begin{align*}ABC\end{align*} | \begin{align*}A(1, 1)\end{align*} | \begin{align*}B(5, 5)\end{align*} | \begin{align*}C(-2, 4)\end{align*} |
Reflected Image \begin{align*}TOM\end{align*} | \begin{align*}T(1, -1)\end{align*} | \begin{align*}O(5, -5)\end{align*} | \begin{align*}M(-2, -4)\end{align*} |
b) The measures of each angle are shown in the diagram.
Congruent angles have the same measure. Since \begin{align*}\angle A=\angle T, \angle B=\angle O\end{align*}, and \begin{align*}\angle C=\angle M\end{align*}, the angles are congruent.
c)
The side lengths are also the same \begin{align*}(m \overline{AC}=m \overline{TM},m \overline{BC}=m \overline{OM},\end{align*} and \begin{align*}m \overline{AB}=m \overline{TO})\end{align*}
d) Since the angle measures are the same and the lengths of each side are the same, the triangles are congruent. Triangle \begin{align*}ABC\end{align*} has been reflected to produce triangle \begin{align*}TOM\end{align*}.
Examples
Example A
Graph the following two squares to determine if Square A is reflected to form Square B.
Square A Vertices: \begin{align*}A(2, 3), B(8, 3), C(8, 8),\end{align*} and \begin{align*}D(2, 8)\end{align*}.
Square B Vertices: \begin{align*}W(-2, 3), X(-8, 3), Y(-8, 8),\end{align*} and \begin{align*}Z(-2, 8)\end{align*}.
The side lengths are the same \begin{align*}(m \overline{AB}=m \overline{WX},m \overline{BC}=m \overline{WZ}, m \overline{CD}=m \overline{YZ}\end{align*} and \begin{align*}m \overline{DA}=m \overline{XY})\end{align*}.
Since the side measures are the same (both shapes are squares and therefore all angles are \begin{align*}90^\circ\end{align*}), Square A is congruent to Square B. Each \begin{align*}x\end{align*}-coordinate on \begin{align*}ABCD\end{align*} is multiplied by –1 to produce Square B. With all of this, you can conclude that Square B is a reflected image of the preimage Square A. The reflection is about the \begin{align*}y\end{align*}-axis.
Example B
Describe how you would know if image \begin{align*}T\end{align*} is reflected onto image \begin{align*}T^\prime\end{align*}.
As indicated on the graph below, the side measures are the same and the angle measures are all the same. Image \begin{align*}T\end{align*} is congruent to Image \begin{align*}T^\prime\end{align*}. Each \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinate is multiplied by –1. With all of this, you can conclude that the Image \begin{align*}T\end{align*} is reflected about the origin to form Image \begin{align*}T^\prime\end{align*}.
Example C
Describe how you would know if image \begin{align*}A\end{align*} is reflected to form image \begin{align*}A^\prime\end{align*}.
As indicated on the graph below, the side measures are the same and the angle measures are all the same. Image \begin{align*}A\end{align*} is congruent to Image \begin{align*}A^\prime\end{align*}. Looking at point \begin{align*}A\end{align*} and \begin{align*}A^\prime\end{align*}, the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates have been reversed. With all of this, you can conclude that Image \begin{align*}A\end{align*} is reflected across the line \begin{align*}y=x\end{align*} to form the image \begin{align*}A^\prime\end{align*}.
Vocabulary
- Congruent Angles
- Congruent angles have the same measure.
- Line of Symmetry
- The line of symmetry is really just a line of reflection but it allows the point's preimage to be reflected to a translated image and maintain their measurements.
- Reflections
- Reflections in transformations involve flipping a shape or figure over a line of reflection, a point of reflection, or a plane of reflection. In reflections, all of the points in the preimage and the translated image are equidistant from this line of reflection.
Guided Practice
1. Describe how you would know if image A is reflected onto image B.
2. Describe how you would know if image S is reflected onto image T.
3. Is \begin{align*}ABCDEF\end{align*} a reflected image of \begin{align*}A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime\end{align*}? How do you know?
Answers
1.
As indicated on the graph, the side measures are the same and the angle measures are all the same. Image A is congruent to Image B. Each \begin{align*}x\end{align*}-coordinate point is multiplied by –1 while each \begin{align*}y\end{align*}-coordinate value remains the same. With all of this, you can conclude that Image B is a reflected image of the preimage Image A. Preimage A is in fact reflected in the \begin{align*}y\end{align*}-axis to form Image B.
2.
As indicated on the graph, the side measures are the same and the angle measures are all the same. Image S is congruent to Image T. Each \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinate has been multiplied by –1. With all of this, you can conclude that Image T is a reflected image of the Preimage S. As well, Preimage A has been reflected across the line \begin{align*}y=x\end{align*} in order to produce Image T.
3.
As indicated on the graph, the side measures are the same and the angle measures are all the same. Image \begin{align*}ABCDEF\end{align*} is congruent to Image \begin{align*}A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime \end{align*}. Each \begin{align*}y\end{align*}-coordinate has been multiplied by –1. With all of this, you can conclude that Image \begin{align*}A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime\end{align*} is a reflected image of the Preimage \begin{align*}ABCDEF\end{align*}. As well, Preimage \begin{align*}ABCDEF\end{align*} has been reflected across the line \begin{align*}y\end{align*}-axis in order to produce Image \begin{align*}A^\prime B^\prime C^\prime D^\prime E^\prime F^\prime\end{align*}.
Summary
The previous three lessons you worked with reflections, a transformation of a figure that involves flipping a point or a figure along the Cartesian plane. In this lesson you looked more closely at the properties of reflections. These properties include:
- line segments are the same length (distance)
- angle measures in a figure remain the same (angles)
- reverse orientation
- points that are on each line remain on the line for the reflected image (collinear)
Through the use of geometry software, it is easy to measure both the angles and the side lengths to determine if these elements are congruent, thus satisfying the first two properties. If points are each moving in the same direction and measure the same distance, you can be sure that the preimage is reflected to form the new image.
Problem Set
Describe how you would know if image A is reflected onto image B in each of the following diagrams.
In each case, is the second image a reflection of preimage A? How do you know?
Summary
In this second lesson of Chapter Is it a Slide, a Flip, or a Turn? you have been introduced to reflections. Remember that a reflection is just a flip of the preimage to create a translated image. Reflections are an important concept as you notice them probably on a daily basis in your everyday life. You look into a mirror and see a reflection of yourself! You can hold up your two hands and one is a reflection of the other. Reflections are also found in the study of chemical compounds only here they use the word chirality to describe the reflection. Chirality simply means that the structures of the compounds have opposite orientation and are, really, just mirror images of each other.
You first learned to describe the reflection using words. So you learned to look at points in a reflected image and its preimage to determine if the images involve:
- a reflection in the \begin{align*}x\end{align*}-axis,
- a reflection in the \begin{align*}y\end{align*}-axis,
- a reflection about the origin, and
- a reflection in the line \begin{align*}y = x\end{align*} or \begin{align*}y = -x\end{align*}.
In this lesson you described the type of reflection using words. Following this, you learned to draw a reflected image based on a description. You further identified the types of reflections using the following data.
Type of Reflection | What happens... |
---|---|
Reflections in the \begin{align*}x\end{align*}-axis | \begin{align*}x\end{align*} values stay the same, \begin{align*}y\end{align*} values multiply by –1 |
Reflections in the \begin{align*}y\end{align*}-axis | \begin{align*}y\end{align*} values stay the same, \begin{align*}x\end{align*} values multiply by –1 |
Reflections about the origin | Both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values multiply by –1 |
Reflections in the line \begin{align*}y = x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image |
Reflections in the line \begin{align*}y = -x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image and multiply them by –1 |
Therefore if you were asked to reflect an image in the line \begin{align*}y=x\end{align*}, you quickly knew to interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates of the preimage to form the translated image.
In the third concept, you learned to write the notation for the reflected image from a preimage. Expanding on the table above, you could graph a reflected image and then write the notation for the image. Your new table to use when working with reflections became:
Line (point) of reflection | Points on preimage | Notation rule |
---|---|---|
\begin{align*}x\end{align*}-axis | \begin{align*}x\end{align*} values stay the same, \begin{align*}y\end{align*} values multiply by –1 | \begin{align*}r_{x-axis}(x,y) \rightarrow (x,-y)\end{align*} |
\begin{align*}y\end{align*}-axis | \begin{align*}y\end{align*} values stay the same, \begin{align*}x\end{align*} values multiply by –1 | \begin{align*}r_{y-axis}(x,y) \rightarrow (-x,y)\end{align*} |
\begin{align*}y = x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image | \begin{align*}r_{y=x}(x,y) \rightarrow (y,x)\end{align*} |
\begin{align*}y = -x\end{align*} | Interchange the \begin{align*}x\end{align*}- and \begin{align*}y\end{align*}-coordinates to form the reflected image and multiply both by –1 | \begin{align*}r_{y=-x}(x,y) \rightarrow (-y,-x)\end{align*} |
Origin: (0, 0) | Both \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values multiply by –1 | \begin{align*}r_{origin}(x,y) \rightarrow (-x,-y)\end{align*} |
Finally in the last concept of this lesson, you learned that reflections have the following properties:
- line segments are the same length (distance)
- angle measures in a figure remain the same (angles)
- reverse orientation
- points that are on each line remain on the line for the reflected image (collinear)
You used these tendencies to prove that one image is a reflection of another image by finding the lengths of the preimage and the reflected image as well as the angle measures in both images. You then found where the points moved from the preimage to the reflected image. All of these properties combined proved that the one image was a reflection of the other.
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