12.5: Composition of Transformations
Learning Objectives
- Perform a glide reflection.
- Perform a reflection over parallel lines and the axes.
- Determine a single transformation that is equivalent to a composite of two transformations.
Review Queue
- Reflect \begin{align*}ABCD\end{align*}
ABCD over the \begin{align*}x-\end{align*}x− axis. Find the coordinates of \begin{align*}A'B'C'D'\end{align*}A′B′C′D′ . - Translate \begin{align*}A'B'C'D'\end{align*}
A′B′C′D′ such that \begin{align*}(x,y) \rightarrow (x+4,y)\end{align*}(x,y)→(x+4,y) . Find the coordinates of \begin{align*}A''B''C''D''\end{align*}A′′B′′C′′D′′ .
Know What? An example of a glide reflection is your own footprint. The equations to find your average footprint are in the diagram to the right. Find your average footprint and write the transformation rule for one stride.
Glide Reflections
Now that we have learned all our rigid transformations, or isometries, we can perform more than one on the same figure.
Composition (of transformations): To perform more than one transformation on a figure.
Glide Reflection: A composition of a reflection and a translation. The translation is in a direction parallel to the line of reflection.
For any glide reflection, order does not matter.
In the Review Queue above, you performed a glide reflection on \begin{align*}ABCD\end{align*}
Example 1: Reflect \begin{align*}\triangle ABC\end{align*} over the \begin{align*}y-\end{align*}axis and then translate the image 8 units down.
Solution: The green image to the right is the final answer.
\begin{align*}A(8,8) & \rightarrow A''(-8,0)\\ B(2,4) & \rightarrow B''(-2,-4)\\ C(10,2) & \rightarrow C''(-10,-6)\end{align*}
Compositions can always be written as one rule.
Example 2: Write a single rule for \begin{align*}\triangle ABC\end{align*} to \begin{align*}\triangle A''B''C''\end{align*} from Example 1.
Solution: Looking at the coordinates of \begin{align*}A\end{align*} to \begin{align*}A''\end{align*}, the \begin{align*}x-\end{align*}value is the opposite sign and the \begin{align*}y-\end{align*}value is \begin{align*}y - 8\end{align*}. Therefore the rule would be \begin{align*}(x,y) \rightarrow (-x, y - 8)\end{align*}.
Reflections over Parallel Lines
The next composition we will discuss is a double reflection over parallel lines. For this composition, we will only use horizontal or vertical lines.
Example 3: Reflect \begin{align*}\triangle ABC\end{align*} over \begin{align*}y = 3\end{align*} and \begin{align*}y = -5\end{align*}.
Solution: Unlike a glide reflection, order matters. Therefore, you would reflect over \begin{align*}y = 3\end{align*} first, (red triangle) then a reflection over \begin{align*}y = -5\end{align*} (green triangle).
Example 4: Write a single rule for \begin{align*}\triangle ABC\end{align*} to \begin{align*}\triangle A''B''C''\end{align*} from Example 3.
Solution: In the graph, the two lines are 8 units apart \begin{align*}(3-(-5)=8)\end{align*}. The figures are 16 units apart. The double reflection is the same as a translation that is double the distance between the parallel lines.
\begin{align*}(x,y) \rightarrow (x,y-16)\end{align*}
Reflections over Parallel Lines Theorem: The composition of two reflections over parallel lines that are \begin{align*}h\end{align*} units apart, it is the same as a translation of \begin{align*}2h\end{align*} units.
Be careful with this theorem because it does not say which direction the translation is in.
Example 5: \begin{align*}\triangle DEF\end{align*} has vertices \begin{align*}D(3, -1), E(8, -3)\end{align*}, and \begin{align*}F(6, 4)\end{align*}. Reflect \begin{align*}\triangle DEF\end{align*} over \begin{align*}x = -5\end{align*} and \begin{align*}x = 1\end{align*}. Determine which one translation this double reflection would be the same as.
Solution: From the Reflections over Parallel Lines Theorem, we know that this double reflection is going to be the same as a single translation of \begin{align*}2(1 -(-5))\end{align*} or 12 units.
First, reflect over \begin{align*}x = -5\end{align*}
Second, reflect over \begin{align*}x = 1\end{align*}
Comparing the preimage and image, this is a translation of 12 units to the right.
If the lines of reflection were switched, then it would have been a translation of 12 units to the left.
Reflections over the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} Axes
You can also reflect over intersecting lines. First, we will reflect over the \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axes.
Example 6: Reflect \begin{align*}\triangle DEF\end{align*} from Example 5 over the \begin{align*}x-\end{align*}axis, followed by the \begin{align*}y-\end{align*}axis. Find the coordinates of \begin{align*}\triangle D''E''F''\end{align*} and the one transformation this double reflection is the same as.
Solution: \begin{align*}\triangle D''E''F''\end{align*} is the green triangle in the graph to the left. If we compare the coordinates of it to \begin{align*}\triangle DEF\end{align*}, we have:
\begin{align*}D(3,-1) & \rightarrow D'(-3,1)\\ E(8,-3) & \rightarrow E'(-8,3)\\ F(6,4) & \rightarrow F'(-6,-4)\end{align*}
From the rules of rotations in the previous section, this is also an \begin{align*}180^\circ\end{align*} rotation.
Reflection over the Axes Theorem: If you compose two reflections over each axis, then the final image is a rotation of \begin{align*}180^\circ\end{align*} of the original.
With this particular composition, order does not matter.
Reflections over Intersecting Lines
For this composition, we are going to take it out of the coordinate plane.
Example 7: Copy the figure below and reflect it over \begin{align*}l\end{align*}, followed by \begin{align*}m\end{align*}.
Solution: The easiest way to reflect the triangle is to fold your paper on each line of reflection and draw the image. It should look like this:
(Patty paper could be used here).
The green triangle is the final answer.
Investigation 12-2: Double Reflection over Intersecting Lines
Tools Needed: Example 7, protractor, ruler, pencil
- Take your answer from Example 7 and measure the angle of intersection for lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*}. If you copied it from the text, it is \begin{align*}55^\circ\end{align*}.
- Draw lines from the corresponding points on the blue triangle and the green triangle.
- Measure this angle using your protractor. How does it related to \begin{align*}55^\circ\end{align*}?
If you copied the image exactly from the text, the angle is \begin{align*}110^\circ\end{align*} counterclockwise.
Notice that order would matter in this composition. If we had reflected the blue triangle over \begin{align*}m\end{align*} followed by \begin{align*}l\end{align*}, then the green triangle would be rotated \begin{align*}110^\circ\end{align*} clockwise.
Reflection over Intersecting Lines Theorem: A composition of two reflections over lines that intersect at \begin{align*}x^\circ\end{align*}, then the resulting image is a rotation of \begin{align*}2x^\circ\end{align*}. The center of rotation is the point of intersection.
Example 8: A square is reflected over two lines that intersect at a \begin{align*}79^\circ\end{align*} angle. What one transformation will this be the same as?
Solution: From the theorem above, this is the same as a rotation of \begin{align*}2 \cdot 79^\circ=178^\circ\end{align*}.
Know What? Revisited The average 6 foot tall man has a \begin{align*}0.415 \times 6 = 2.5\end{align*} foot stride. Therefore, the transformation rule for this person would be \begin{align*}(x,y) \rightarrow (-x,y+2.5)\end{align*}.
Review Questions
- Questions 1-3 use the theorems learned in this section.
- Questions 4-12 are similar to Examples 1 and 2.
- Questions 13-19 are similar to Examples 3-5.
- Questions 20-22 are similar to Example 6.
- Questions 23-30 are similar to Example 8 and use the theorems learned in this section.
- Explain why the composition of two or more isometries must also be an isometry.
- What one transformation is the same as a reflection over two parallel lines?
- What one transformation is the same as a reflection over two intersecting lines?
Use the graph of the square to the left to answer questions 4-6.
- Perform a glide reflection over the \begin{align*}x-\end{align*}axis and to the right 6 units. Write the new coordinates.
- What is the rule for this glide reflection?
- What glide reflection would move the image back to the preimage?
Use the graph of the square to the left to answer questions 7-9.
- Perform a glide reflection to the right 6 units, then over the \begin{align*}x-\end{align*}axis. Write the new coordinates.
- What is the rule for this glide reflection?
- Is the rule in #8 different than the rule in #5? Why or why not?
Use the graph of the triangle to the left to answer questions 10-12.
- Perform a glide reflection over the \begin{align*}y-\end{align*}axis and down 5 units. Write the new coordinates.
- What is the rule for this glide reflection?
- What glide reflection would move the image back to the preimage?
Use the graph of the triangle to the left to answer questions 13-15.
- Reflect the preimage over \begin{align*}y = -1\end{align*} followed by \begin{align*}y = -7\end{align*}. Draw the new triangle.
- What one transformation is this double reflection the same as?
- Write the rule.
Use the graph of the triangle to the left to answer questions 16-18.
- Reflect the preimage over \begin{align*}y = -7\end{align*} followed by \begin{align*}y = -1\end{align*}. Draw the new triangle.
- What one transformation is this double reflection the same as?
- Write the rule.
- How do the final triangles in #13 and #16 differ?
Use the trapezoid in the graph to the left to answer questions 20-22.
- Reflect the preimage over the \begin{align*}x-\end{align*}axis then the \begin{align*}y-\end{align*}axis. Draw the new trapezoid.
- Now, start over. Reflect the trapezoid over the \begin{align*}y-\end{align*}axis then the \begin{align*}x-\end{align*}axis. Draw this trapezoid.
- Are the final trapezoids from #20 and #21 different? Why do you think that is?
Answer the questions below. Be as specific as you can.
- Two parallel lines are 7 units apart. If you reflect a figure over both how far apart with the preimage and final image be?
- After a double reflection over parallel lines, a preimage and its image are 28 units apart. How far apart are the parallel lines?
- Two lines intersect at a \begin{align*}165^\circ\end{align*} angle. If a figure is reflected over both lines, how far apart will the preimage and image be?
- What is the center of rotation for #25?
- Two lines intersect at an \begin{align*}83^\circ\end{align*} angle. If a figure is reflected over both lines, how far apart will the preimage and image be?
- A preimage and its image are \begin{align*}244^\circ\end{align*} apart. If the preimage was reflected over two intersecting lines, at what angle did they intersect?
- A preimage and its image are \begin{align*}98^\circ\end{align*} apart. If the preimage was reflected over two intersecting lines, at what angle did they intersect?
- After a double reflection over parallel lines, a preimage and its image are 62 units apart. How far apart are the parallel lines?
Review Queue Answers
- \begin{align*}A'(-2, -8), B'(4, -5), C'(-4, -1), D'(-6, -6)\end{align*}
- \begin{align*}A''(2, -8), B''(8, -5), C''(0, -1), D''(-2, -6)\end{align*}