Explanation:
a) Reflection in the line y = x:
Invariance under reflection in the line y = x means that the vertices remain the same after the reflection. To determine the invariant vertices, we need to find the points on the line y = x.
The line y = x passes through points (0, 0), (1, 1), (2, 2), and so on. All the points on this line are invariant under reflection in the line y = x.
Therefore, the invariant vertices are:
(0, 0), (1, 1), (2, 2), ...
b) Reflection in the line y = x followed by a 180-degree rotation about (5, 5):
First, we need to find the points that are invariant under reflection in the line y = x. We can follow the same process as in part a) to find the invariant vertices.
Next, we apply a 180-degree rotation about the point (5, 5) to the invariant vertices obtained from the reflection. The rotation will maintain the distance and orientation of the points relative to the center of rotation.
To find the vertices that remain invariant after the reflection and rotation, we need to identify the points that are equidistant from the line y = x and the point (5, 5).
The points equidistant from the line y = x and (5, 5) are:
(5, 5) and (0, 10)
Therefore, the invariant vertices under the given transformation are:
(0, 0), (1, 1), (2, 2), ..., (5, 5), (0, 10)