Here are a few recommended readings before getting started with this lesson.
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.
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.
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.
Reflections can be performed by hand with the help of a straightedge and a compass.
To reflect across the line follow these three steps.
To reflect and repeat the two previous steps.
The image of after the reflection is the triangle formed by and
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.
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.
The line of reflection is the perpendicular bisector of any segment connecting a point to its image.
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.
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.
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.
Draw both reflections over the original photo. Is there a single transformation that maps Kevin’s house onto the final image?
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.
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.
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.
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.
In the diagram above, was first reflected across and then reflected across
Additionally, there is also a theorem for the case where the lines of reflection intersect each other.
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
Let be the intersection point between the lines and To prove that is a rotation of the following two facts will be proved.
By definition of reflection, the line is the perpendicular bisector of the segments and
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
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.
Factor out
To this point, a few compositions of rigid motions have been analyzed.
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.
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
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
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 |
Start by plotting the point along with the line and the vector
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!