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Congruence, Proof, and Constructions

Reflections of Figures in a Plane

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.

Explore

Reflecting a Triangle

In the following applet, the vertices of can be moved. Also, the slope of the line can be set by moving the slider point. Once everything is set, reflect across line
Is there any relationship between and If so, do and have the same relationship? What about and

Discussion

Properties of a Reflection

As can be checked in the previous exploration, after a point is reflected across a line the segment connecting with its image is perpendicular to Additionally, the line intersects at its midpoint.
A point and its reflection across a line
From these two facts, it can be concluded that is the perpendicular bisector of With this information in mind, reflections can be defined properly.

Discussion

Definition of a Reflection

Concept

Reflection

A reflection is a transformation in which every point of a figure is reflected in a line. The line in which the points are reflected is called the line of reflection and acts like a mirror.
Triangle being reflected across a movable line
More precisely, a reflection across a line maps every point in the plane into its image such that one of the following statements is satisfied.
  • If is on the line then and are the same point.
  • If is not on the line then is the perpendicular bisector of
Segment AA' and line ell showing the right angle and the congruence hatches
Like rotations and translations, reflections are also rigid motions because they preserve the side lengths and angle measures. However, reflections change the orientation of the preimage.

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.

A quadrilateral and a line

Perform a reflection to the chemistry lab across the walkway in order to draw the physics lab.

Answer

Quadrilateral ABCD and its image under the reflection across ell

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.

Quadrilateral ABCD and line ell

To reflect across a reflection can be performed on each vertex, one at a time. For example, to reflect is a good start. To do so, follow the definition of reflections. First, draw a line perpendicular to passing through

Line perpendicular to ell passing through A

Then can be plotted as the point on line where its distance to is the same as the distance from to

Image of A over the line m

The same steps can be applied to reflect vertices and Notice that because both and are on the line their images will maintain their same point locations, respectively.

Image of A, B, C, and D

Finally, the image of under a reflection across the line is the quadrilateral formed by and This quadrilateral represents the physics lab.

Quadrilateral ABCD and its image under the reflection across ell

Discussion

Constructing a Reflection

Reflections can be performed by hand with the help of a straightedge and a compass.

Triangle ABC and line ell

To reflect across the line follow these three steps.

1

Draw an Arc Centered at that Intersects at Two Points
Place the compass tip on vertex and draw an arc that intersects the line at two different points. Let and be these points.
Drawing an Arc centered at A that intersects ell at two points

2

Draw Two Arcs Centered at the Intersection Points
With the same compass setting, place the compass tip on and draw an arc. Keeping the same setting, place the compass tip on and draw a second arc. The intersection point of these two arcs is the image of
Drawing arcs centered at P and Q that intersect each other
Notice that both arcs need to be drawn on the side of not containing vertex

3

Repeat the Previous Steps for the Other Vertices

To reflect and repeat the two previous steps.

Images of A, B, and C

The image of after the reflection is the triangle formed by and

Triangle ABC and triangle A'B'C'
Notice that the first two steps are the same as the first two steps to construct a line perpendicular to a given line through a point.

Discussion

Reflections Across Certain Lines

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 and Investigate each relationship by using the following applet.
Applet to investigate the coordinates of a point after a reflection across the coordinate axes and the lines y=x and y=-x
Drawn from diagram, the following relations can be determined.
  • The image of under a reflection across the axis is
  • The image of under a reflection across the axis is
  • The image of under a reflection across the line is
  • The image of under a reflection across the line is

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 want's to know for sure.

A painting showing a pentagon and its reflection an unknown certain line

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

A painting showing a pentagon, a line of reflection and the reflection

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

A painting showing a pentagon and its reflection an unknown certain line

Next, construct the perpendicular bisector of That will represent the line of reflection used to make the painting.

A painting showing a pentagon, a line of reflection and the reflection

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.

A painting showing a pentagon and its reflection an unknown certain line

Pop Quiz

Practicing Reflections

In the following applet, there are two possible requests.

To reflect place points and 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.

Performing random reflections to random triangles

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.

A photo of Kevin's house and two 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?

Answer

Hint

When a figure is reflected, its orientation is flipped. What happens if the figure is reflected a second time? Draw the segments connecting the vertices of Kevin's house to the corresponding vertices of the final image. Look for a relationship between these segments.

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.

Drawing lines Perp. to the midline through each vertex

The image of after the reflection is a point on such that and are equidistant from the midline. The images of and can be located in a similar way. By connecting and the image of Kevin's house after performing the first reflection can be obtained.

Drawing the reflected vertices

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 and have the same orientation, in contrast to and
  • The corresponding sides of and are parallel.
  • The vectors and are all parallel, congruent, and have the same direction.
Notice that the third statement corresponds to the definition of translations. Consequently, can be mapped onto using a single transformation, a translation.
Translating polygon ABCD onto A''B''C''D''

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.

Reflection of a triangle across two parallel lines

In the diagram, is first reflected across and then the image is reflected across Equivalently, the following statements hold true.

  • Triangle is a translation of triangle along
  • Vector is perpendicular to and
  • The magnitude of is where is the distance between and

Proof

The proof will be developed focusing the attention on vertex and its images, but the conclusions are true for all the vertices. First, start reflecting across By definition of reflection, the line is the perpendicular bisector of Let be the intersection point of and

Reflection of a triangle across two parallel lines

Next, perform the reflection of across line This time, the line is the perpendicular bisector of Let be the intersection point of and

Reflection of a triangle across two parallel lines

Since and are parallel lines and is perpendicular to by the Perpendicular Transversal Theorem, is perpendicular to Also, is perpendicular to Consequently, and are parallel vectors with a common point. Therefore, these vectors belong to the same line.

The points and are collinear.

Due to collinearity, it can be concluded that is perpendicular to and In addition, by the Segment Addition Postulate, the length of can be rewritten in terms of the lengths of and Remember that, by definition of reflection, and Substitute these values into the previous equation.
Substitute values and evaluate

Segment Addition Postulate

Finally, notice that is the distance between the lines, that is, That way, it has been shown that is perpendicular to and and is twice the distance between the lines.

Generalizing

Applying the same reasoning it can be concluded that and are perpendicular to and From these relations, it is obtained that and are parallel segments. Additionally, all these segments have the same length. Last but not least, notice that and all have the same direction, which is the same as pointing from the first line to the second line. Consequently, is a translation of along where is either or
Reflection of a triangle across two parallel lines

Extra

Different Positions of and
Notice that the vertices of the preimage can be positioned differently relatively to the lines of reflection. Here, the proof was developed for positioned not too far to the left of which resulted in lying between the two lines. However, there are several other possibilities.
Reflection of a triangle across two parallel lines
The same is true for vertices and 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.
Reflection of a triangle across two parallel lines

In the diagram above, was first reflected across and then reflected across

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.

Reflection of a triangle across two intersecting lines

In the diagram, is first reflected across and then the image is reflected across The same result is obtained when is rotated by an angle of about point

Proof

Let be the intersection point between the lines and To prove that is a rotation of the following two facts will be proved.

  • The point is the same distance from a vertex of as it is from the corresponding vertex of That is, and
  • The angles and have all a measure of where is the acute angle formed by the lines of reflection.

Distance From to Each Vertex

By definition of reflection, the line is the perpendicular bisector of the segments and

Reflection of triangle ABC across line ell along with the segments AA', BB', and CC'

Since is on the Perpendicular Bisector Theorem guarantees that is equidistant from the endpoints of and Similarly, the line is the perpendicular bisector of and Therefore, is equidistant from the endpoints of these segments.

Reflection of triangle A'B'C' across line m along with the segments A'A'', B'B'', and C'C''

In consequence, and Finally, the Transitive Property of Equality can be used to obtain the first part of the proof.

Flowchart showing that PA=PA'', PB=PB'', PC=PC''

That way, it has been shown that is the same distance from a vertex of as it is from the corresponding vertex of

Angle of Rotation

Here, it will be shown that is where is the acute angle formed by the lines. Let and be the intersection points between and and and respectively.

Drawing segments AA' and A'A'' along with points Q and T

Next, consider the right triangles and Notice that their hypotenuses and are congruent as well as their legs and

Drawing Triangles PQA and PQA'

Therefore, and are congruent thanks to the Hypotenuse Leg Theorem. This congruence implies that and have the same measure.

Similarly, by the Hypotenuse Leg Theorem, and are congruent. Consequently, and are congruent.

Now, applying the Angle Addition Postulate, can be rewritten as the sum of and 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).
Substitute for and simplify
Substitute for and simplify
Finally, notice that is the angle formed by the lines of reflection, that is, Consequently, the measure of is twice the measure of the angle formed by the lines.

Applying a similar reasoning, it can be shown that and are also equal to This completes the proof of the fact that is a rotation of
Rotating Triangle ABC about P to map it to Triangle A''B''C''

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. Since a glide reflection is a composition of rigid motions, it is also a rigid motion.
Performing a Glide Reflection on a Triangle
Thanks to the fact that the line of reflection and the translation vector are parallel, a glide reflection could instead be a reflection followed by a translation. That is, the image does not depend on the order of the transformations.
Performing a Glide Reflection on a Triangle

Example

Performing a Glide Reflection

Consider a glide reflection defined by the line and the vector Apply this glide reflection to What are the coordinates of

Hint

The order in which the transformations are applied does not affect the image. When a point is reflected across the line the image has coordinates

Solution

Algebraic Solution

The given glide reflection is the composition of a translation along and a reflection across Start by plotting the point along with the line and the vector

Point P, line y=-x and vector v

Remember, the order in which the transformations are applied does not affect the image. To begin, the translation can be performed first. To translate along add to the coordinate of and to the coordinate of

Point Image

Next, reflect the point across the line To do that, the coordinates are swapped and their signs are changed.

Point Image
Consequently, the coordinates of are

Solution

Geometric Solution

Start by plotting the point along with the line and the vector

Point P, line y=-x and vector v
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 draw the vector such that is the tail of the vector. The tip of the vector is the translation of
Drawing vector v
Next, reflect the obtained image across To do it, draw the line perpendicular to passing through Then, is the point on this line such that and are equidistant from
Drawing vector v
Consequently, the coordinates of are

Closure

Reflections as Rigid Motions

Although there are four types of rigid motionsrotations, 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.
Flowchart showing that any rigid motion is the composition of reflections
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!

Mountain reflected on a lake
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