6. Reflections of Figures in a Plane
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Chapter 1
6.
Reflections of Figures in a Plane
| Student Learning Objectives: |
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Why
Reflections in geometry are transformations where figures are flipped over a line, producing a mirror image. Glide reflections combine translations (slides) and reflections, resulting in a figure's shift and flip. When figures are reflected over intersecting lines, the combined effect can be described as a single rotation. This means the figure is turned around a point without changing its shape or size. Understanding these transformations is crucial in various fields, from design to engineering, as they provide insights into how shapes and patterns can be manipulated and repositioned within a given space.
Lesson Settings & Tools
| | 15 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Reflections of Figures in a Plane
Slide of 15
Apart from rotations and translations, there is a third type of rigid motion called reflections. In a plane, reflections act similar to mirrors. However, unlike a mirror, a figure is reflected across a line. In this lesson, the formal definition and properties of reflections will be developed.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Reflecting a Triangle
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Discussion
Properties of a Reflection
As can be checked in the previous exploration, after a point A is reflected across a line l, the segment connecting A with its image is perpendicular to l. Additionally, the line l intersects AA' at its midpoint.
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Discussion
Reflections
A reflection is a transformation in which every point of a figure is reflected across a line. The line across the points are reflected in what is called the line of reflection. This acts like a mirror.
- If A is on the line l, then A and A' are the same point.
- If A is not on the line l, then l is the perpendicular bisector of AA'.
Like rotations and translations, reflections are rigid motions because they preserve the side lengths and angle measures. However, reflections can change the orientation of the preimage.
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Example
Reflecting a Polygon
The principal of Jefferson High wants to build a physics lab by the chemistry lab. The plan, seen from the sky, is that the new building looks like a reflection of the chemistry lab through the walkway that connects the soccer field with the library.
Perform a reflection to the chemistry lab across the walkway in order to draw the physics lab.
Answer
Hint
To reflect the chemistry lab, reflect each corner of the building. The physics lab is the quadrilateral formed by the images. Remember that the image of a point that is on the line of reflection is the same point.
Solution
For simplicity, start by labeling each corner of the quadrilateral and the walkway.
To reflect ABCD across l, a reflection can be performed on each vertex, one at a time. For example, to reflect A is a good start. To do so, follow the definition of reflections. First, draw a line perpendicular to l passing through A.
Then A' can be plotted as the point on line m where its distance to l is the same as the distance from A to l.
The same steps can be applied to reflect vertices B, C, and D. Notice that because both C and D are on the line l, their images will maintain their same point locations, respectively.
Finally, the image of ABCD under a reflection across the line l is the quadrilateral formed by A', B', C', and D'. This quadrilateral represents the physics lab.
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Discussion
Reflections Performed by Hand
Reflections can be performed by hand with the help of a straightedge and a compass.
To reflect △ ABC across the line l, follow these three steps.
1
Draw an Arc Centered at A that Intersects l at Two Points
expand_more 
2
Draw Two Arcs Centered at the Intersection Points
expand_more With the same compass setting, place the compass tip on P and draw an arc. Keeping the same setting, place the compass tip on Q and draw a second arc. The intersection point of these two arcs is the image of A.
Notice that both arcs need to be drawn on the side of l not containing vertex A.
3
Repeat the Previous Steps for the Other Vertices
expand_more To reflect B and C, repeat the two previous steps.
The image of △ ABC after the reflection is the triangle formed by A', B', and C'.
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Discussion
Reflections in the Coordinate Plane
In the coordinate plane, there is a particular relationship between the coordinates of a point and those of its image after a reflection across certain lines worth considering. These lines are the coordinate axes and lines y=x and y=- x. Investigate each relationship by using the following applet.
- The image of (a,b) under a reflection across the x-axis is (a,- b).
- The image of (a,b) under a reflection across the y-axis is (- a,b).
- The image of (a,b) under a reflection across the line y=x is (b,a).
- The image of (a,b) under a reflection across the line y=- x is (- b,- a).
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Example
Identifying the Line of Reflection
Up to this point, how to perform a reflection, when already given the line of reflection, has been understood. Now, consider a case if given a figure and its image under a reflection. How can the line of reflection be found? This question is answered in the following example.
While visiting a museum, Tearrik saw a painting containing the word MATH
and two pentagons. The picture caught his attention. There is some sort of reflection but he wants to know for sure.
Tearrik analyzed the painting. He determined that the picture was made by performing a reflection. Show how Tearrik figured that out and draw the line of reflection used in the picture.
Answer
Hint
The line of reflection is the perpendicular bisector of any segment connecting a point to its image.
Solution
When a point is reflected across a line, its image is such that the line of reflection is the perpendicular bisector of the segment connecting the point to its image. Therefore, to find the line of reflection of the painting, start by drawing a segment that connects a vertex and its image. For instance, draw CC'.
Next, construct the perpendicular bisector of CC'. That will represent the line of reflection used to make the painting.
Notice that drawing only one segment that connects a point to its image is enough to find the line of reflection. Before performing the reflection, the painting looked as follows.
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Pop Quiz
Practicing Reflections
In the following applet, there are two possible requests.
- Reflect △ ABC across the given line.
- Draw the line of reflection used to map △ ABC onto △ A'B'C'.
To reflect △ ABC, place points A', B', and C' where they should be after the reflection is applied. To draw the line of reflection, place the two points, so they lie on the line of reflection.
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Example
Performing Reflections
In previous lessons, the composition of rotations and translations were studied. Now it is time to learn about the composition of reflections. The first case to consider is when the lines of reflection are parallel.
Using a drone, Kevin took a photo of the roof of his house, Main Street, and Euclid Sreet — they are parallel streets.
Perform the following transformations to Kevin's house.
- Reflect Kevin's house across the midline of Main Street.
- Reflect the obtained image across the midline of Euclid Street.
Draw both reflections over the original photo. Is there a single transformation that maps Kevin's house onto the final image?
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Answer
Solution
To reflect Kevin's house across the midline of Main Street, start by drawing lines perpendicular to the line of reflection passing through each vertex of the house. Label each vertex and line for added clarity.
The image of A after the reflection is a point A' on l_2 such that A' and A are equidistant from the midline. The images of B, C, and D can be located in a similar way. By connecting A', B', C' and D', the image of Kevin's house after performing the first reflection can be obtained.
Applying a similar reasoning, the second reflection can be performed. After the second reflection is performed, the image of Kevin's house lies to the right of Euclid Street.
To determine whether there is a single transformation that maps Kevin's house to the final image, label the vertices of the initial polygon and its images.
From the previous diagram, the following conclusions can be drawn.
- Both ABCD and A''B''C''D'' have the same orientation, in contrast to ABCD and A'B'C'D'.
- The corresponding sides of ABCD and A''B''C''D'' are parallel.
- The vectors AA'', BB'', CC'', and DD'' are all parallel, congruent, and have the same direction.
Notice that the third statement corresponds to the definition of translations. Consequently, ABCD can be mapped onto A''B''C''D'' using a single transformation, a translation.
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Discussion
Reflections Across Parallel Lines
In the previous example, it was concluded that the composition of reflections in parallel lines gives the same result as a translation. This conclusion is not an isolated fact. Actually, there is a theorem that guarantees this result.
Rule
Reflections in Parallel Lines Theorem
The composition of two reflections across parallel lines is a single translation. Furthermore, the translation vector is perpendicular to both parallel lines, and its magnitude is twice the distance between the parallel lines.
In the diagram, △ ABC is first reflected across l and then the image is reflected across m. Equivalently, the following statements hold true.
- Triangle A''B''C'' is a translation of triangle ABC along v.
- Vector v is perpendicular to l and m.
- The magnitude of v is 2d, where d is the distance between l and m.
Proof
The proof will be developed focusing the attention on vertex C and its images, but the conclusions are true for all the vertices. First, start reflecting △ ABC across l. By definition of reflection, the line l is the perpendicular bisector of CC'. Let P_1 be the intersection point of l and CC'.
Next, perform the reflection of △ A'B'C' across line m. This time, the line m is the perpendicular bisector of C'C''. Let P_2 be the intersection point of m and C'C''.
Since l and m are parallel lines and CC' is perpendicular to l, by the Perpendicular Transversal Theorem, CC' is perpendicular to m. Also, C'C'' is perpendicular to m. Consequently, CC' and C'C'' are parallel vectors with a common point. Therefore, these vectors belong to the same line.
The points C, C', and C'' are collinear.
Due to collinearity, it can be concluded that CC'' is perpendicular to l and m. In addition, by the Segment Addition Postulate, the length of CC'' can be rewritten in terms of the lengths of CC' and C'C''. CC'' &= CC' + C'C'' &= (CP_1+P_1C')+(C'P_2+P_2C'') Remember that, by definition of reflection, CP_1=P_1C' and C'P_2=P_2C''. Substitute these values into the equation.
CC'' = CP_1+P_1C'+C'P_2+P_2C''
▼
Substitute values and evaluate
SubstituteII
CP_1= P_1C', P_2C''= C'P_2
CC'' = P_1C'+P_1C'+C'P_2+ C'P_2
AddTerms
Add terms
CC'' = 2P_1C'+2C'P_2
FactorOut
Factor out 2
CC'' = 2(P_1C'+C'P_2)
Segment Addition Postulate
CC'' = 2 P_1P_2
Finally, notice that P_1P_2 is the distance between the lines, that is, P_1P_2=d. That way, it has been shown that CC'' is perpendicular to l and m and CC'' is twice the distance between the lines.
Generalizing
Applying the same reasoning it can be concluded that AA'' and BB'' are perpendicular to l and m. rcr AA''⊥ l &and& AA''⊥ m BB''⊥ l &and& BB''⊥ m CC''⊥ l &and& CC''⊥ m From these relations, it is obtained that AA'', BB'', and CC'' are parallel segments. Additionally, all these segments have the same length. AA'' &= 2d BB'' &= 2d CC'' &= 2d Last but not least, notice that AA'', BB'', and CC'' all have the same direction, which is the same as pointing from the first line to the second line. Consequently, △ A''B''C'' is a translation of △ ABC along v, where v is either AA'', BB'', or CC''.
Extra
Different Positions of C, C', and C''Notice that the vertices of the preimage can be positioned differently relatively to the lines of reflection. Here, the proof was developed for C positioned not too far to the left of l, which resulted in C' lying between the two lines. However, there are several other possibilities.
The same is true for vertices A and B. Nevertheless, the claim is true for any case. One interesting fact here is that the translation vector always points in the same direction as pointing from the first line to the second line.
In the diagram above, △ ABC was first reflected across m and then reflected across l.
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Discussion
Reflections Across Intersecting Lines
Additionally, there is also a theorem for the case where the lines of reflection intersect each other.
Rule
Reflections in Intersecting Lines Theorem
The composition of two reflections across intersecting lines is a single rotation. Additionally, the center of rotation is the point of intersection of the lines, and the angle of rotation is twice the measure of the acute or right angle formed by the lines.
In the diagram, △ ABC is first reflected across l and then the image is reflected across m. The same result is obtained when △ ABC is rotated by an angle of 2α^(∘) about point P.
Proof
Let P be the intersection point between the lines l and m. To prove that △ A''B''C'' is a rotation of △ ABC, the following two facts will be proved.
- The point P is the same distance from a vertex of △ ABC as it is from the corresponding vertex of △ A''B''C''. That is, PA=PA'', PB=PB'', and PC=PC''.
- The angles ∠ APA'', ∠ BPB'', and ∠ CPC'' have all a measure of 2α, where α is the acute angle formed by the lines of reflection.
Distance From P to Each Vertex
By definition of reflection, the line l is the perpendicular bisector of the segments AA', BB', and CC'.
Since P is on l, the Perpendicular Bisector Theorem guarantees that P is equidistant from the endpoints of AA', BB', and CC'. PA &= PA' PB &= PB' PC &= PC' Similarly, the line m is the perpendicular bisector of A'A'', B'B'', and C'C''. Therefore, P is equidistant from the endpoints of these segments.
In consequence, PA'=PA'', PB'=PB'', and PC'=PC''. Finally, the Transitive Property of Equality can be used to obtain the first part of the proof.
That way, it has been shown that P is the same distance from a vertex of △ ABC as it is from the corresponding vertex of △ A''B''C''.
Angle of Rotation
Here, it will be shown that m∠ APA'' is 2α, where α is the acute angle formed by the lines. Let Q and T be the intersection points between AA' and l, and A'A'' and m respectively.
Next, consider the right triangles PQA and PQA'. Notice that their hypotenuses PA and PA' are congruent as well as their legs AQ and A'Q.
Therefore, △ PQA and △ PQA' are congruent thanks to the Hypotenuse Leg Theorem. This congruence implies that ∠ APQ and ∠ QPA' have the same measure.
m∠ APQ = m∠ QPA' (I)
Similarly, by the Hypotenuse Leg Theorem, △ PTA' and △ PTA'' are congruent. Consequently, ∠ A'PT and ∠ TPA'' are congruent.
m∠ A'PT = m∠ TPA'' (II)
Now, applying the Angle Addition Postulate, m∠ APA'' can be rewritten as the sum of m∠ APA' and m∠ A'PA''. m∠ APA'' = m∠ APA' + m∠ A'PA'' Once more, thanks to the Angle Addition Postulate, each of the two angles on the right-hand side can be rewritten in terms of the angle measures involved in Equations (I) and (II).
m∠ APA'' = m∠ APA' + m∠ A'PA''
SubstituteII
m∠ APA'= m∠ APQ+m∠ QPA', m∠ A'PA''= m∠ A'PT+m∠ TPA''
m∠ APA'' = m∠ APQ+m∠ QPA' + m∠ A'PT+m∠ TPA''
▼
Substitute m∠ QPA' for m∠ APQ and simplify
m∠ APA'' = 2m∠ QPA'+ m∠ A'PT+m∠ TPA''
▼
Substitute m∠ A'PT for m∠ TPA'' and simplify
m∠ APA'' = 2m∠ QPA'+ 2m∠ A'PT
FactorOut
Factor out 2
m∠ APA'' = 2(m∠ QPA'+ m∠ A'PT)
Finally, notice that m∠ QPA'+ m∠ A'PT is the angle formed by the lines of reflection, that is, m∠ QPA'+ m∠ A'PT=α. Consequently, the measure of ∠ APA'' is twice the measure of the angle formed by the lines.
m∠ APA'' = 2α
Applying a similar reasoning, it can be shown that m∠ BPB'' and m∠ CPC'' are also equal to 2α. This completes the proof of the fact that △ A''B''C'' is a rotation of △ ABC.
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Discussion
Combining Translations and Reflections
To this point, a few compositions of rigid motions have been analyzed.
- The composition of rotations is a rotation or a translation.
- The composition of translations is a translation.
- The composition of reflections is a translation or a rotation.
In these cases, the composition of two rigid motions can be presented as a single transformation. However, not every composition of two rigid motions can be expressed as a single transformation. Such is the case for glide reflections.
Concept
Glide Reflection
A glide reflection is a transformation that combines a translation and a reflection across a line parallel to the translation vector. It is a composition of rigid motions — meaning it, too, is considered a rigid motion.
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Example
Performing a Glide Reflection
Consider a glide reflection defined by the line y=- x and the vector v=⟨ 2,-2⟩. Apply this glide reflection to P(-4,1). What are the coordinates of P'?
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Hint
The order in which the transformations are applied does not affect the image. When a point (a,b) is reflected across the line y=- x, the image has coordinates (- b,- a).
Solution
Algebraic SolutionThe given glide reflection is the composition of a translation along v and a reflection across y=- x. Start by plotting the point P along with the line y=- x and the vector v=⟨ 2,-2⟩.
Remember, the order in which the transformations are applied does not affect the image. To begin, the translation can be performed first. To translate P along v=⟨ 2,-2⟩, add 2 to the x-coordinate of P and -2 to the y-coordinate of P.
| Point | (a,b)→ (a+v_1,b+v_2) | Image |
|---|---|---|
| P(-4,1) | (-4,1) → (-4+2,1-2) | (-2,-1) |
Next, reflect the point (-2,-1) across the line y=- x. To do that, the coordinates are swapped and their signs are changed.
| Point | (a,b)→(- b,- a) | Image |
|---|---|---|
| (-2,-1) | (-2,-1) → (-(-1),-(-2)) | (1,2) |
Consequently, the coordinates of P' are (1,2).
Solution
Geometric SolutionStart by plotting the point P along with the line y=- x and the vector v=⟨ 2,-2⟩.
Since the order in which the transformations defining the glide reflection are applied does not affect the image, the translation can be performed first. To translate P, draw the vector v=⟨ 2,-2⟩ such that P is the tail of the vector. The tip of the vector is the translation of P.
Next, reflect the obtained image across y=- x. To do it, draw the line perpendicular to y=- x passing through (-2,-1). Then, P' is the point on this line such that (-2,-1) and P' are equidistant from y=- x.
Consequently, the coordinates of P' are (1,2).
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Closure
Reflections as Rigid Motions
Although there are four types of rigid motions — rotations, translations, reflections, and glide reflections — any rigid motion can be seen as a composition of some reflections. That is the case due to the two theorems previously mentioned in this lesson. Interact with the applet to review each rigid motion.
- Since translations and rotations keep the orientation of a figure, two reflections are needed.
- Since the glide reflections combine translations and reflections, three reflections are needed.
In general, any rigid motion is the composition of either one, two, or three reflections. Before moving on from this lesson, keep in mind that reflections are everywhere, so look around and identify some reflections. No, not just the one in the mirror. Think about reflections in nature or a favorite hobby!
External credits: Ethan Conley
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Reflections of Figures in a Plane
Exercise 2.1
In the coordinate plane, we see two triangles, △ ABC and △ A''B''C''.
To move △ ABC to △ A''B''C'', the following glide reflection was performed. Translation:& (x,y) → (x+ ,y+ ) Reflection:& across the line y=x Write down the correct translation.
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From the exercise we know that △ A''B''C'' is created by reflecting its preimage across y=x. To perform a reflection across y=x, we use the following rule. Reflection Across the Line y=x (x,y) → (y,x) We can use the same rule to undo the reflection and find the coordinates of the vertices of △ A'B'C'.
| △ A''B''C'' | △ A'B'C' |
|---|---|
| A''(5,6) | A'(6,5) |
| B''(4,2) | B'(2,4) |
| C''(-1,5) | C'(5,-1) |
Let's graph △ A'B'C' together with △ A''B''C''.
To determine what translation that takes △ ABC to △ A'B'C', we will now compare the corresponding vertices of the triangles.
As we can see, the triangle ABC is translated 4 units to the right and 4 units up.
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Solution
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To reflect △ XYZ across line k, we must draw lines that are perpendicular to k and that pass through each of the vertices. Since both k and m are vertical lines, the perpendicular lines will be horizontal. Let's remove the different options for now and draw these lines.
To perform the first reflection, we move each vertex of △ XYZ to the opposite side of k such that they are at the same distance from k.
Now we must perform a second reflection across line m. By reinstating the four options, we can determine which one is correct.
As we can see, B is the correct option.
To determine the translation which carries △ XYZ onto B, we must measure the distance between corresponding vertices.
We have a translation of 12 units to the right.
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At which point, K through O, should you hit the 8-ball E, using the cue ball C, in order for it to go into the upper-middle pocket?
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To begin, we will remove the different options to have a clearer vision of the path needed. Let's draw the path the ball can travel that will bounce off the sidewall and then into the upper-middle pocket.
To determine the correct path, we can reflect the 8-ball across the wall. Then draw a path from the reflected 8-ball to the upper-middle pocket. The reflection's path should align with the path drawn on the pool table.
Let's bring back the original five options and check which point the path goes through.
We should hit the 8-ball so it bounces off point M.
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Reflections of Figures in a Plane
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