We are asked to imagine a situation where we reflect a triangle in Quadrant I over the y-axis and then translate the image 2 units left and 3 units down. We want to know whether there is a single transformation that maps the preimage onto the image. Let's consider an example triangle in Quadrant I.
Let's try to find a single transformation that maps the preimage to the image. So far, we know two types of transformations: translations and reflections.
Translations
First, note that the orientation of the two triangles are different.
A translation does not change the orientation of a figure, so we can not map â–³ ABC onto â–³ A''B''C'' with a single translation. Let's now check whether we can use a single reflection to map â–³ ABC onto â–³ A''B''C''.
Reflections
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.
Considering each pair of vertices gives us a different line of reflection. Therefore, there is no single reflection that maps â–³ ABC to â–³ A''B''C''.
