6. Reflections of Figures in a Plane
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Chapter 1
6.
Reflections of Figures in a Plane
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.
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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
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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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Reflecting a Triangle
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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.
From these two facts, it can be concluded that l is the perpendicular bisector of AA'. With this information in mind, reflections can be defined properly.
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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.
In more precise terms, a reflection across a line l maps every point A in the plane onto its image A' such that one of the following statements is satisfied. 
- 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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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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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
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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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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.
Drawn from diagram, the following relations can be determined.
- 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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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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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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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.
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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.
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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.
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
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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Reflections Across Intersecting Lines
Additionally, there is also a theorem for the case where the lines of reflection intersect each other.
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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)
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)
m∠ APA'' = 2α
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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.
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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. 
The line of reflection and the translation vector are parallel. That means a glide reflection can be a reflection followed by a translation, rather than only a translation followed by a reflection. The image does not depend on the order of the transformations. 
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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'? 
Consequently, the coordinates of P' are (1,2).
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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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) |
Solution
Geometric SolutionStart by plotting the point P along with the line y=- x and the vector v=⟨ 2,-2⟩.
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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.
Therefore, there is no need for more transformations other than reflections when talking about rigid motions. The minimum number of reflections that need to be composed can be determined by looking at the orientation of a figure and its image closely. 
- 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
Reflections of Figures in a Plane
Exercises
Consider △ ABC in the coordinate plane.
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Let's start by drawing the line of reflection. This is a vertical line through x=10. We will also identify the coordinates of each vertex in the triangle.
To perform the reflection, each vertex of △ ABC must be moved along a perpendicular line to the line of reflection until it ends up on the opposite side and at the same distance from x=10 as its corresponding vertex on △ ABC.
Finding Perpendicular Lines
Notice that x=10 is a vertical line. This means any perpendicular line must be a horizontal line. Let's draw them in the diagram.
Reflecting the Triangle
To reflect the triangle, we must find the distance from A, B, and C to the line of reflection. Since all vertices are located at lattice points, we can determine these. Distance from A: 12-10=2 Distance from B: 17-10=7 Distance from C: 15-10=5 When we know these distances, we can mark the reflected vertices and draw △ A'B'C'.
As we can see, A' has the coordinates (8,6).
As in Part A, we will start by drawing the line of reflection. This time, we have a horizontal line of reflection.
Finding Perpendicular Lines
As in Part A, we must find perpendicular lines to the line of reflection. Since the line of reflection is horizontal, any perpendicular line must be vertical. Let's draw them through each of the triangle's vertices.
Reflecting the Triangle
Again, like in Part A, we will find the distance from A, B, and C to the line of reflection by calculating the vertical distance between the vertices and the line of reflection. Distance from A: 8-6=2 Distance from B: 8-5=3 Distance from C: 8-3=5 Now we can reflect the triangle.
As we can see, C' has the coordinates (15,13).
Let's first graph the line of reflection.
Finding Perpendicular Lines
To determine perpendicular lines to this line of reflection is not as straightforward as in Part A and B. However, we can do it with the use of a compass. Let's remove the grid lines and labels for now.
By using a compass, we can find the correct distance from the line of reflection.
If we repeat this procedure for A and B, we can plot the reflected triangle.
As we can see, B' is roughly at (13,12).
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A triangle has vertices in D(1,3), E(4,4), and F(3,1).
Which of the following options is a reflection of △ DEF across the x-axis?
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Which of the following options is a reflection of △ DEF across the y-axis?
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Which of the following options is a reflection of △ DEF across y=x.
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Which of the following options is a reflection of △ DEF across y=- x.
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Let's plot the triangle in the coordinate plane.
When we reflect a point across the x-axis the x-coordinate stays the same while the y-coordinate changes sign. Reflection Across the $x$-axis (x,y) → (x,- y) Using this rule, we can find the x- and y-coordinates of the vertices of the image.
| △ DEF | (x,y) | (x, - y) |
|---|---|---|
| D | (1,3) | (1,-3) |
| E | (4,4) | (4,-4) |
| F | (3,1) | (3,-1) |
With this information, we can reflect the triangle across the x-axis. Let's also add the four options and check which one the reflection maps onto.
B is the correct option.
When a point is reflected across the y-axis the y-coordinate is unchanged while the x-coordinate changes sign. Reflection Across the $y$-axis (x,y) → (- x,y) Using this rule, we can find the x- and y-coordinates of the reflection's vertices.
| △ DEF | (x,y) | (- x,y) |
|---|---|---|
| D | (1,3) | (- 1,3) |
| E | (4,4) | (-4,4) |
| F | (3,1) | (-3,1) |
Now we can reflect the triangle across the y-axis. If we add the different options, we can select which one the reflection maps onto.
C is the correct option.
When a point is reflected across the line y=x, the x- and y-coordinate switch positions. Reflection Across the Line y=x (x,y) → (y,x) Let's use this rule to determine the x- and y-coordinates of the reflection's vertices.
| △ DEF | (x,y) | (y,x) |
|---|---|---|
| D | (1,3) | (3,1) |
| E | (4,4) | (4,4) |
| F | (3,1) | (1,3) |
Now we can draw the reflected triangle and match it with the correct option.
B is the correct option.
Reflecting a point across the line y=- x, switches the point's x- and y-coordinate. Additionally, the sign of both coordinates switch. Reflection Across the Line y=- x (x,y) → (- y,- x) Using the rule, we can find the x- and y-coordinates of the reflection.
| △ DEF | (x,y) | (- y,- x) |
|---|---|---|
| D | (1,3) | (- 3,- 1) |
| E | (4,4) | (- 4,- 4) |
| F | (3,1) | (- 1,- 3) |
Now we can draw the reflected triangle and match it with the correct option.
B is the correct option.
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Point A has the coordinates (4,1). What are the coordinates of A'' after point A undergoes the following glide reflection.
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Let's plot the point in a coordinate plane.
First we will perform the translation. The rule tells us to move the point one step down.
Next, we want to reflect the point across the y-axis. To do that, we will use the following rule. Reflection Across they-axis (x,y) → (- x,y) Let's use the rule on point A' to find the coordinates of A''.
The point has the coordinates (-4,0).
Again, we will begin by plotting the point in a coordinate plane.
This time, we must perform a horizontal translation by moving the point three steps to the left.
Next, we want to reflect the point across the x-axis. To do that, we will use the following rule. Reflection Across the $x$-axis (x,y) → (x,- y) Let's reflect the point across the x-axis.
The point has the coordinates (1,-1).
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Consider the following triangle and translation vector.
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To perform the transformation, we must move the translation vector until its starting point maps onto each triangle's vertices. At the end of the vectors, we find the translated point.
When something is reflected across the line y=- x, we use the following rule. Reflection Across the Line y=- x (x,y) → (- y,- x) Using the rule, we can find the x- and y-coordinates of the reflection.
| △ A'B'C' | (x,y) | (- y,- x) |
|---|---|---|
| A' | (-3,0) | (0,3) |
| B' | (-3,-2) | (2,3) |
| C' | (0,-2) | (2,0) |
Now we can draw the reflected triangle.
The coordinates of A'' is (0,3).
A glide reflection is a combination of a translation and a reflection where the line of reflection is parallel to the translation vector. For glide reflections, the final position of the image does not change according to the order of the transformations. Let's have a look at the translation vector and line of reflection.
As we can see, the translation vector and line of reflection are not parallel. In fact in this particular case, they are perpendicular. Therefore, this does not represent a glide reflection. If we were to perform the combination of transformations in the reverse order, we would not get the same image as in Part A.
This does not match △ A''B''C'' from Part A. Therefore, it is not a glide reflection.
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The triangle ABC is reflected across a line on the form y=ax forming triangle A'B'C'.
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We know that the line of reflection is on the form y=ax, which means it passes through the origin.
To determine the line of reflection, we must find one more point through which the line passes. To find that point, we remember that the line of reflection is the perpendicular bisector to the segment that can be drawn between two corresponding vertices. Let's draw AA'.
A perpendicular bisector is a line which is perpendicular to a segment and that passes through the segment's midpoint. To find the midpoint, we will use a compass. Place the needle point of the compass at either end of the segment and draw two arcs with the same radius across the segment. Make the arcs' radius greater than half the length of the segment.
We can find the perpendicular bisector by placing a straightedge in-line with the arcs' points of intersection and drawing a line through them. This line divides AA' into two congruent halves at a right angle.
The perpendicular bisector intersects AA' at (8,4).
Great work so far. We can now use the Slope Formula to calculate the slope.
The slope is 12, which means the line of reflection can be written as y= 12x.
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What angle of rotation maps X onto X''?
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When a polygon is reflected across two intersecting lines, the reflections can be described by a single rotation. This is formalized by the Reflections in Intersecting Lines Theorem.
Reflections in Intersecting Lines Theorem |- The combination of two reflections across intersecting lines can be described by a single rotation.
The angle of rotation has a measure that is twice the size of the angle between the intersecting lines. From the diagram, we see that the angle between the intersecting lines is 50^(∘). Therefore, the angle of rotation that maps X onto X'' must be 100^(∘).
This time, we see that the angle between the intersecting lines is 20^(∘). This means the angle of rotation that carries X onto X'' is 40^(∘).
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Reflections of Figures in a Plane
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