Final answer:
To determine if the given polygons are congruent, we examined the possibility of a transformation consisting of a 90-degree rotation and reflection, followed by a translation. Applying the rule (x, y) → (-y + 3, x - 3), we find that the polygons are congruent as each vertex maps exactly onto the corresponding vertex of the other polygon.
Step-by-step explanation:
Determining Polygon Congruence through Transformation
When determining whether two polygons are congruent, we compare their shapes and sizes, without considering their positions. The given vertices for two polygons are A(2, -1), B(3, 0), C(2, 3), and A'(1, 2), Β΄(0, 3), C'(-3, 2). To check if these polygons are congruent, we can attempt to find a transformation that maps one polygon onto the other.
A congruence transformation could include translation, rotation, and/or reflection. We are looking for a rule of the form (x, y) → (x', y') where x' and y' express the new coordinates after applying the transformation.
Congruence check involves three steps:
Find the distance between the corresponding vertices of the two polygons.
Compare the slopes of the sides to ensure the angles are the same.
Check for any possible translations, rotations, or reflections.
For our polygons, we notice a consistent vertical and horizontal shift between corresponding points when we examine the coordinates. However, considering distance and slope alone it's not enough because the orientation of the polygons after such simple translation is not maintained.
In detail, a counterclockwise rotation of 90 degrees about the origin and then a reflection over the line y = -x followed by a translation can precisely map the vertices of the first polygon to those of the second one. Here is the transformation rule: (x, y) → (-y, x). Then, by translating, (x, y) → (-y + 3, x - 3).
Applying this transformation, A becomes A'', B becomes B'', and C becomes C'', mapping exactly to A', B', and C', respectively, thus indicating that the two polygons are congruent.