The homothetic geometry on C is (C,H) where H is the group of scaling of C. Which figures are congruent in (C,H) to the circle of radius 1 centered at 1? How could you describe the congruence class in language understandable by a high school geometry student?
In homothetic geometry, a circle of radius r centered at point (a, b) is defined as the set of all points (x, y) such that the distance between (x, y) and (a, b) is equal to r.
In this case, we have the circle of radius 1 centered at 1. So, the circle is the set of all points (x, y) such that the distance between (x, y) and (1, 0) is equal to 1.
To find which figures are congruent to this circle in (C, H), we need to consider the group of scalings, H, in Homothetic geometry.
The group of scalings, H, in this case, consists of all transformations that can be achieved by scaling, but not by translating or rotating points.
In (C, H), the only figure that is congruent to the circle of radius 1 centered at 1 is another circle. This is because the group of scalings H allows for transformations that change the size of the figure, but not its shape or position. Since a circle can only be transformed into another circle by scaling, no other figure can be congruent to the given circle in (C, H).
To describe this congruence class in language understandable by a high school geometry student, you can say that all circles of any size centered at any point on the complex plane (C) are congruent to each other in (C, H). This means that if you take a circle and scale it up or down in size (while keeping its shape), it will still be considered congruent to the original circle.