- #1
Zoey93
- 15
- 0
Here is the question:
Provide a logical argument that demonstrates that when applying any two reflections, the outcome will always either be equivalent to a rotation or a translation.
This is what I came up with:
•Translation
A reflection is defined as, “a transformation in which a geometric figure is reflected across a line, creating a mirror image. The line is called the axis of reflection” (uCertify). So, in order for a transformation to be a reflection there must be a line or lines representing the axis of reflection. We know that lines in a plane must either intersect or be parallel. Reflecting a figure in a line and then reflecting the image in a parallel line has the same result as translating the figure in a direction perpendicular to the reflection lines for a distance equal to twice the distance between the lines. In other words, the product of two reflections across parallel lines will always produce a translation.
•Rotation
A reflection is defined as, “a transformation in which a geometric figure is reflected across a line, creating a mirror image. The line is called the axis of reflection” (uCertify). The line is called the axis of reflection” (uCertify). So, in order for a transformation to be a reflection there must be a line or lines representing the axis of reflection. We know that lines in a plane must either intersect or be parallel. Reflecting a figure in a line and then reflecting the image in an intersecting line has the same results as rotating the original figure about the intersection point of the lines by an angle equal to twice the angle formed by the reflection lines. In other words, the product of two reflections across perpendicular lines will always produce a rotation.
My professor told me that I need more explanation for why two reflections across parallel lines results in a translation and why two reflections across intersecting lines results in a rotation. He said that I just seem to say it is that way and that I give properties for a rotation and translation, but that I don't explain why two reflections result in a rotation or translation.
Can anyone please help me figure this out??
Provide a logical argument that demonstrates that when applying any two reflections, the outcome will always either be equivalent to a rotation or a translation.
This is what I came up with:
•Translation
A reflection is defined as, “a transformation in which a geometric figure is reflected across a line, creating a mirror image. The line is called the axis of reflection” (uCertify). So, in order for a transformation to be a reflection there must be a line or lines representing the axis of reflection. We know that lines in a plane must either intersect or be parallel. Reflecting a figure in a line and then reflecting the image in a parallel line has the same result as translating the figure in a direction perpendicular to the reflection lines for a distance equal to twice the distance between the lines. In other words, the product of two reflections across parallel lines will always produce a translation.
•Rotation
A reflection is defined as, “a transformation in which a geometric figure is reflected across a line, creating a mirror image. The line is called the axis of reflection” (uCertify). The line is called the axis of reflection” (uCertify). So, in order for a transformation to be a reflection there must be a line or lines representing the axis of reflection. We know that lines in a plane must either intersect or be parallel. Reflecting a figure in a line and then reflecting the image in an intersecting line has the same results as rotating the original figure about the intersection point of the lines by an angle equal to twice the angle formed by the reflection lines. In other words, the product of two reflections across perpendicular lines will always produce a rotation.
My professor told me that I need more explanation for why two reflections across parallel lines results in a translation and why two reflections across intersecting lines results in a rotation. He said that I just seem to say it is that way and that I give properties for a rotation and translation, but that I don't explain why two reflections result in a rotation or translation.
Can anyone please help me figure this out??
