In a reflection, each point of the preimage and its image are the same distance from the line of reflection. To find a translation that maps the preimage onto the image, compare the coordinates of the vertices of the preimage and the coordinates of the corresponding vertices of the image.
No, see solution.
Practice makes perfect
We are asked to consider a situation where we reflect a nonregular figure over the x-axis and then reflect it over the y-axis. Let's consider an example figure and perform these transformations.
We want to know whether there is a single transformation using reflections or translations that maps the preimage onto the image. First, let's try to find a reflection that maps our preimage onto our image.
Reflection
In a reflection, each point of the preimage and its image are the same distance from the line of reflection. Then, the line of reflection must go through the midpoints of the segments connecting each vertex of the preimage with its image. A reflection is a mirror image, so the line of reflection must be perpendicular to these segments.
Each pair of vertices gives us a different line of reflection. Therefore, there is no single reflection that maps the preimage onto the image. Now, let's try to find a translation!
Translation
To find a translation that maps our preimage onto the image, let's first note the coordinates of the vertices of the preimage and the coordinates of the corresponding vertices of the image.
We can see that each vertex of the preimage needs to be translated to the left and down to be mapped onto the corresponding vertex of the image. Let's recall how we can describe a translation a units left and b units down using symbols.
(x, y) → (x - a, y - b)
Finally, let's find a translation that maps each vertex of the preimage onto the corresponding vertex of the image.
Preimage
Translation
Image
A(2,2)
(2,2) → (2 - 4, 2 - 4)
A''(- 2, - 2)
B(4,1)
(4,1) → (4 - 8, 1 - 2)
B''(- 4, - 1)
C(1,4)
(1,4) → (1 - 2, 4 - 8)
C''(- 1, - 4)
As we can see, mapping each vertex to its image requires a different translation. Therefore there is no single translation that maps the original figure onto the image of a reflection over the x-axis followed by a reflection over the y-axis.
