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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.

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.
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

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 is called the line of reflection and acts like a mirror.
More precisely, a reflection across a line maps every point in the plane onto its image such that one of the following statements is satisfied.

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.

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.

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 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

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

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.

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

Discussion

Reflections Performed by Hand

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

To reflect across the line follow these three steps.

1
Draw an Arc Centered at that Intersects at Two Points
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Place the compass tip on vertex and draw an arc that intersects the line at two different points. Let and be these points.
2
Draw Two Arcs Centered at the Intersection Points
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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
Notice that both arcs need to be drawn on the side of not containing vertex
3
Repeat the Previous Steps for the Other Vertices
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To reflect and repeat the two previous steps.

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

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 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 and Investigate each relationship by using the following applet.
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 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.

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

Next, construct the perpendicular bisector of 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.

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.

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?

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.

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.

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.
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, 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

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

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 equation.
Substitute values and evaluate

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

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.
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.

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.

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

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.

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

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.

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

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).