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.
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CK-12 Foundation: Chapter12ReflectionsA
Watch more about transformations and isometries by watching 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 \begin{align*}A\end{align*}, then the image would be labeled @$\begin{align*}A'\end{align*}@$, said “a prime.” If there is an image of @$\begin{align*}A'\end{align*}@$, that would be labeled @$\begin{align*}A''\end{align*}@$, 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 @$\begin{align*}y-\end{align*}@$axis: If @$\begin{align*}(x,y)\end{align*}@$ is reflected over the @$\begin{align*}y-\end{align*}@$axis, then the image is @$\begin{align*}(-x,y)\end{align*}@$.
- Reflection over the @$\begin{align*}x-\end{align*}@$axis: If @$\begin{align*}(x,y)\end{align*}@$ is reflected over the @$\begin{align*}x-\end{align*}@$axis, then the image is @$\begin{align*}(x,-y)\end{align*}@$.
- Reflection over @$\begin{align*}x = a:\end{align*}@$ If @$\begin{align*}(x, y)\end{align*}@$ is reflected over the vertical line @$\begin{align*}x = a\end{align*}@$, then the image is @$\begin{align*}(2a - x, y)\end{align*}@$.
- Reflection over @$\begin{align*}y = b:\end{align*}@$ If @$\begin{align*}(x, y)\end{align*}@$ is reflected over the horizontal line @$\begin{align*}y = b\end{align*}@$, then the image is @$\begin{align*}(x, 2b - y)\end{align*}@$.
- Reflection over @$\begin{align*}y = x\end{align*}@$: If @$\begin{align*}(x, y)\end{align*}@$ is reflected over the line @$\begin{align*}y = x\end{align*}@$, then the image is @$\begin{align*}(y, x)\end{align*}@$.
- Reflection over @$\begin{align*}y = -x\end{align*}@$: If @$\begin{align*}(x, y)\end{align*}@$ is reflected over the line @$\begin{align*}y = -x\end{align*}@$, then the image is @$\begin{align*}(-y, -x)\end{align*}@$.
Example A
Reflect the letter ‘‘F’’ over the @$\begin{align*}x-\end{align*}@$axis.
To reflect the letter @$\begin{align*}F\end{align*}@$ over the @$\begin{align*}x-\end{align*}@$axis, now the @$\begin{align*}x-\end{align*}@$coordinates will remain the same and the @$\begin{align*}y-\end{align*}@$coordinates will be the same distance away from the @$\begin{align*}x-\end{align*}@$axis on the other side.
Example B
Reflect @$\begin{align*}\triangle ABC\end{align*}@$ over the @$\begin{align*}y-\end{align*}@$axis. Find the coordinates of the image.
To reflect @$\begin{align*}\triangle ABC\end{align*}@$ over the @$\begin{align*}y-\end{align*}@$axis the @$\begin{align*}y-\end{align*}@$coordinates will remain the same. The @$\begin{align*}x-\end{align*}@$coordinates will be the same distance away from the @$\begin{align*}y-\end{align*}@$axis, but on the other side of the @$\begin{align*}y-\end{align*}@$axis.
@$$\begin{align*}A(4,3) & \rightarrow A'(-4,3)\\ B(7,-1) & \rightarrow B'(-7,-1)\\ C(2,-2) & \rightarrow C'(-2,-2)\end{align*}@$$
Example C
Reflect the triangle @$\begin{align*}\triangle ABC\end{align*}@$ with vertices @$\begin{align*}A(4, 5), B(7, 1)\end{align*}@$ and @$\begin{align*}C(9, 6)\end{align*}@$ over the line @$\begin{align*}x = 5\end{align*}@$.
Notice that this vertical line is through our preimage. Therefore, the image’s vertices are the same distance away from @$\begin{align*}x = 5\end{align*}@$ as the preimage. As with reflecting over the @$\begin{align*}y-\end{align*}@$axis (or @$\begin{align*}x = 0\end{align*}@$), the @$\begin{align*}y-\end{align*}@$coordinates will stay the same.
@$$\begin{align*}A(4,5) & \rightarrow A'(6,5)\\ B(7,1) & \rightarrow B'(3,1)\\ C(9,6) & \rightarrow C'(1,6)\end{align*}@$$
Example D
Reflect square @$\begin{align*}ABCD\end{align*}@$ over the line @$\begin{align*}y = x\end{align*}@$.
The purple line is @$\begin{align*}y = x\end{align*}@$. To reflect an image over a line that is not vertical or horizontal, you can fold the graph on the line of reflection.
@$$\begin{align*}A(-1, 5) & \rightarrow A'(5, -1)\\ B(0, 2) & \rightarrow B'(2, 0)\\ C(-3, 1) & \rightarrow C'(1, -3)\\ D(-4, 4) & \rightarrow D'(4, -4)\end{align*}@$$
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter12ReflectionsB
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 @$\begin{align*}x-\end{align*}@$axis.
Guided Practice
1. Reflect the line segment @$\begin{align*}\overline{PQ}\end{align*}@$ with endpoints @$\begin{align*}P(-1, 5)\end{align*}@$ and @$\begin{align*}Q(7, 8)\end{align*}@$ over the line @$\begin{align*}y = 5\end{align*}@$.
2. A triangle @$\begin{align*}\triangle LMN\end{align*}@$ and its reflection, @$\begin{align*}\triangle L'M'N'\end{align*}@$ are to the left. What is the line of reflection?
3. Reflect the trapezoid TRAP over the line @$\begin{align*}y = -x\end{align*}@$.
Answers:
1. Here, the line of reflection is on @$\begin{align*}P\end{align*}@$, which means @$\begin{align*}P'\end{align*}@$ has the same coordinates. @$\begin{align*}Q'\end{align*}@$ has the same @$\begin{align*}x-\end{align*}@$coordinate as @$\begin{align*}Q\end{align*}@$ and is the same distance away from @$\begin{align*}y = 5\end{align*}@$, but on the other side.
@$$\begin{align*}P(-1,5) & \rightarrow P'(-1,5)\\ Q(7,8) & \rightarrow Q'(7,2)\end{align*}@$$
2. Looking at the graph, we see that the preimage and image intersect when @$\begin{align*}y = 1\end{align*}@$. Therefore, this is the line of reflection.
3. The purple line is @$\begin{align*}y = -x\end{align*}@$. You can reflect the trapezoid over this line just like we did in Example D.
@$$\begin{align*}T(2,2) & \rightarrow T'(-2, -2)\\ R(4, 3) & \rightarrow R'(-3,-4)\\ A(5, 1) & \rightarrow A'(-1,-5)\\ P(1, -1) & \rightarrow P'(1,-1)\end{align*}@$$
Explore More
- Which letter is a reflection over a vertical line of the letter @$\begin{align*}b\end{align*}@$?
- Which letter is a reflection over a horizontal line of the letter @$\begin{align*}b\end{align*}@$?
Reflect each shape over the given line.
- @$\begin{align*}y-\end{align*}@$axis
- @$\begin{align*}x-\end{align*}@$axis
- @$\begin{align*}y = 3\end{align*}@$
- @$\begin{align*}x = -1\end{align*}@$
Find the line of reflection of the blue triangle (preimage) and the red triangle (image).
Two Reflections The vertices of @$\begin{align*}\triangle ABC\end{align*}@$ are @$\begin{align*}A(-5, 1), B(-3, 6)\end{align*}@$, and @$\begin{align*}C(2, 3)\end{align*}@$. Use this information to answer questions 10-13.
- Plot @$\begin{align*}\triangle ABC\end{align*}@$ on the coordinate plane.
- Reflect @$\begin{align*}\triangle ABC\end{align*}@$ over @$\begin{align*}y = 1\end{align*}@$. Find the coordinates of @$\begin{align*}\triangle A'B'C'\end{align*}@$.
- Reflect @$\begin{align*}\triangle A'B'C'\end{align*}@$ over @$\begin{align*}y = -3\end{align*}@$. Find the coordinates of @$\begin{align*}\triangle A''B''C''\end{align*}@$.
- What one transformation would be the same as this double reflection?
Two Reflections The vertices of @$\begin{align*}\triangle DEF\end{align*}@$ are @$\begin{align*}D(6, -2), E(8, -4)\end{align*}@$, and @$\begin{align*}F(3, -7)\end{align*}@$. Use this information to answer questions 14-17.
- Plot @$\begin{align*}\triangle DEF\end{align*}@$ on the coordinate plane.
- Reflect @$\begin{align*}\triangle DEF\end{align*}@$ over @$\begin{align*}x = 2\end{align*}@$. Find the coordinates of @$\begin{align*}\triangle D'E'F'\end{align*}@$.
- Reflect @$\begin{align*}\triangle D'E'F'\end{align*}@$ over @$\begin{align*}x = -4\end{align*}@$. Find the coordinates of @$\begin{align*}\triangle D''E''F''\end{align*}@$.
- What one transformation would be the same as this double reflection?
Two Reflections The vertices of @$\begin{align*}\triangle GHI\end{align*}@$ are @$\begin{align*}G(1, 1), H(5, 1)\end{align*}@$, and @$\begin{align*}I(5, 4)\end{align*}@$. Use this information to answer questions 18-21.
- Plot @$\begin{align*}\triangle GHI\end{align*}@$ on the coordinate plane.
- Reflect @$\begin{align*}\triangle GHI\end{align*}@$ over the @$\begin{align*}x-\end{align*}@$axis. Find the coordinates of @$\begin{align*}\triangle G'H'I'\end{align*}@$.
- Reflect @$\begin{align*}\triangle G'H'I'\end{align*}@$ over the @$\begin{align*}y-\end{align*}@$axis. Find the coordinates of @$\begin{align*}\triangle G''H''I''\end{align*}@$.
- What one transformation would be the same as this double reflection?