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# 12.3: Reflections

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Reflect a figure over a given line.
• Find the rules for reflections.

## Review Queue

1. Define reflection in your own words.
2. Plot \begin{align*}A(-3, 2)\end{align*}. Translate \begin{align*}A\end{align*} such that \begin{align*}(x,y) \rightarrow (x+6,y)\end{align*}.
3. What line is halfway between \begin{align*}A\end{align*} and \begin{align*}A’\end{align*}?

Know What? A lake can act like a mirror in nature. Describe the line of reflection in the picture to the right.

## Reflections over an Axis

Reflection: A transformation that turns a figure into its mirror image by flipping it over a line.

Line of Reflection: The line that a figure is reflected over.

Example 1: Reflect \begin{align*}\triangle ABC\end{align*} over the \begin{align*}y-\end{align*}axis. Find the coordinates of the image.

Solution: \begin{align*}\triangle A’B’C’\end{align*} will be the same distance away from the \begin{align*}y-\end{align*}axis as \begin{align*}\triangle ABC\end{align*}, but on the other side.

\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*}

From this example, we can generalize a rule for reflecting a figure over the \begin{align*}y-\end{align*}axis.

Reflection over the \begin{align*}y-\end{align*}axis: \begin{align*}(x,y) \rightarrow (-x,y)\end{align*}

Example 2: Reflect the letter \begin{align*}“F”\end{align*} over the \begin{align*}x-\end{align*}axis.

Solution: To reflect the letter \begin{align*}F\end{align*} over the \begin{align*}x-\end{align*}axis, the \begin{align*}y-\end{align*}coordinates will be the same distance away from the \begin{align*}x-\end{align*}axis, but on the other side of the \begin{align*}x-\end{align*}axis.

Reflection over the \begin{align*}x-\end{align*}axis: \begin{align*}(x,y) \rightarrow (x,-y)\end{align*}

## Reflections over Horizontal and Vertical Lines

We can also reflect a figure over any vertical or horizontal line.

Example 3: 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*}.

Solution: The image’s vertices are the same distance away from \begin{align*}x = 5\end{align*} as the preimage.

\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 4: 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*}.

Solution: The line of refection is on \begin{align*}P\end{align*}, which means \begin{align*}P’\end{align*} has the same coordinates. \begin{align*}Q’\end{align*} 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*}

From these examples we have learned that if a point is on the line of reflection then the image is the same as the preimage.

Example 5: 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?

Solution: 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.

If the image does not intersect the preimage, find the midpoint between the preimage and its image. This point is on the line of reflection.

Reflections over \begin{align*}y = x\end{align*} and \begin{align*}y = -x\end{align*}

Example 6: Reflect square \begin{align*}ABCD\end{align*} over the line \begin{align*}y = x\end{align*}.

Solution: The purple line is \begin{align*}y = x\end{align*}. 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*}

From this example, we see that the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values are switched.

Reflection over \begin{align*}y = x\end{align*}: \begin{align*}(x,y)\rightarrow(y,x)\end{align*}

Example 7: Reflect the trapezoid \begin{align*}TRAP\end{align*} over the line \begin{align*}y = -x\end{align*}.

Solution: The purple line is \begin{align*}y = -x\end{align*}. You can reflect the trapezoid over this line just like we did in Example 6.

\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*}

From this example, we see that the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} values are switched with the opposite signs.

Reflection over \begin{align*}y = -x\end{align*}: \begin{align*}(x,y) \rightarrow (-y,-x)\end{align*}

From all of these examples, we notice that a reflection is an isometry.

Know What? Revisited The white line in the picture is the line of reflection. This line coincides with the water’s edge.

## Review Questions

• Questions 1-5 are similar to Examples 1, 3, 4, 6, and 7.
• Questions 6 and 7 are similar to Example 2.
• Questions 8-19 are similar to Examples 1, 3, 4, 6, and 7.
• Questions 20-22 are similar to Example 5.
• Questions 23-30 are similar to Examples 3 and 4.
1. If (5, 3) is reflected over the \begin{align*}y-\end{align*}axis, what is the image?
2. If (5, 3) is reflected over the \begin{align*}x-\end{align*}axis, what is the image?
3. If (5, 3) is reflected over \begin{align*}y = x\end{align*}, what is the image?
4. If (5, 3) is reflected over \begin{align*}y = -x\end{align*}, what is the image?
5. Plot the four images. What shape do they make? Be specific.
6. Which letter is a reflection over a vertical line of the letter \begin{align*}“b”\end{align*}?
7. Which letter is a reflection over a horizontal line of the letter \begin{align*}“b”\end{align*}?

Reflect each shape over the given line.

1. \begin{align*}y-\end{align*}axis
2. \begin{align*}x-\end{align*}axis
3. \begin{align*}y = 3\end{align*}
4. \begin{align*}x = -1\end{align*}
5. \begin{align*}x-\end{align*}axis
6. \begin{align*}y-\end{align*}axis
7. \begin{align*}y = x\end{align*}
8. \begin{align*}y = -x\end{align*}
9. \begin{align*}x = 2\end{align*}
10. \begin{align*}y = -4\end{align*}
11. \begin{align*}y = -x\end{align*}
12. \begin{align*}y = x\end{align*}

Find the line of reflection 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 23-26.

1. Plot \begin{align*}\triangle ABC\end{align*} on the coordinate plane.
2. 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*}.
3. 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*}.
4. 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 27-30.

1. Plot \begin{align*}\triangle DEF\end{align*} on the coordinate plane.
2. 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*}.
3. 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*}.
4. What one transformation would be the same as this double reflection?

1. Examples are: To flip an image over a line; A mirror image.
2. \begin{align*}A'(3, 2)\end{align*}
3. the \begin{align*}y-\end{align*}axis

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Date Created:
Feb 22, 2012