Final answer:
After a rotation transformation, a shape's congruence is preserved but its orientation may change, unless the rotation is 360 degrees. The unchanged congruence is due to distances between points remaining constant during rotation, and orientation changes relative to the axis and angle of rotation.
Step-by-step explanation:
When a shape undergoes a rotation transformation, its congruence and orientation change depending on the angle and the axis of the rotation. The congruence of a shape, which means its size and shape remains unchanged, is preserved after the rotation. However, the orientation of the shape can change unless the rotation is 360 degrees or a full multiple thereof. For example, a square rotated 90 degrees around its center will maintain congruence, as distances between points remain the same, but its orientation will change relative to its original position.
Different shapes have different rotational symmetries based on their geometry. A cube, for instance, has four-fold rotational axes (C4 axes) and three-fold rotational axes (C3 axes). The location of the shape's center of mass is crucial as it affects the rotational inertia and angular momentum of the shape.
Mathematically, rotation in two dimensions is represented by coordinate transformations. A point with coordinates (x, y) in a coordinate system S can be expressed in a rotated system S' by x' = x cos φ + y sin φ and y' = -x sin φ + y cos φ. This describes the conservation of vector magnitude under rotations of the coordinate system, ensuring the congruence of a shape is preserved.