Composition of Reflections Theorems
Parallel Lines, Intersecting Lines, Translations
Theorems involving reflections in mathematics
Note: Usually mathematicians have a keen sense of orderliness and consistency. However, there is an exception to this habit of consistency when it comes to the
conventions/notation for compositions of transformations. There are two separate ways of interpreting the following symbols:
The one convention is to read the composition from right to left
ie. First perform ry-axis and then perform rx-axis
However, an equally prevelant notation is to read from left to right
ie. First perform rx-axis and then perform ry-axis
So what are you to do?
If you are reading a math book or learning from a teacher,either source has, hopefully, already indicated which notation you are following. If you happen to be in the state of New York, the Math B Regents exam follows the right to left notation
, a notation that is consistent with that of composition of functions
In order to accomodate both notations for compositions of reflections, many of the exercises on this page have radio buttons that allow you to chose 'left to right' or 'right to left'. However, the bottom exercises all use the 'right to left' notation that New York State and others use
The shape on the left is going to be reflected across the line y =1,
then the image will be reflected across y = 2,
then the new image will be reflected across the line y = 3,
and so on and so forth across y= 4, y = 5, y = 6, y = 7 …..y = 101.
After all of the reflections have occurred is the transformation from the pre-image(shown on the left) to the final image a translation? Why or why not?
A reflection across 2 parallel lines or across 4 parallel lines or across any even number of parallel lines is equivlant to some kind of translation.
Lets look at the pattern:
||Is it a translation at this time?
||y = 1
||No, 1 refletion is not a translation
||yes, 2 reflections across parallel lines is a translation
||y = 3
||No, 3 reflections is not equivalent to a translation
|| y = 4
||Yes, 4 reflections across parallel lines is a translation
||y = 100
||yes, 100 is an even number and a reflection across an even number of parallel lines is a translation.
Intersecting Lines Theorem
A composition of reflections over intersecting lines is a rotation.
Given the triangle below, perform a composition of reflections over the x-axis then the y-axis, then determine how to
express that composition of reflections
as a mathematical rotation
(Right to Left)
The composition of reflections over the x-axis then the line y=x is equivalent to what rotation about the origin.
The composition of reflections over the y-axis then x-axis is equivalent to what rotation about the origin.
Perform Composition of reflections
What rotation expresses the composition of reflections.
Perform the compostion of reflection then determine what rotation expresses the composition.
Restate the composition as a rotation.
Imagine that a computer program is going to take the point (3,1) and perform the composition of reflections of
a total of 100 times! What are the coordinates of the final image?
What would the coordinates of the image be if the program only performed 99 compositions of the reflection? (Helpful hint: You will arrive at the same answer to this problem if you read the compostion right to left or left to right)
Remember : rx-axis• ry-axis is a rotation by 180° about the origin. Two of these compositions of reflections therefore equals a rotation by 360° (2•180=360)
around the origin− which puts the point back in the exact same spot! Therefore, since 100 is a multiple of 2, the final image after 100 compositions is the same as the pre-image: (3,1). However, if the program only performs the composition of reflections 99 times then this composition is equivalen to a single rotation by 180 around the origin, and the image will be (−3,−1)