symmetry and rotations of parallelograms and rhombus?

Question

1. What are the symmetries of a nonsquare rectangle?
2. What are the symmetries of a parallelogram that is neither a rectangle or a rhombus.
3. What are the symmetries of a rhombus that is not a rectangle?

Answer 1

1. Set up the coordinate system, such that the origin is at the center of the rectangle. x-axis parallels to one pair of sides and y-axis parallels to the other pair. The symmetries are:
a) 180 degree rotational symmetry around z-axis.
b) 180 degree rotational symmetry around x-axis.
c) 180 degree rotational symmetry around y-axis.
d) reflective symmetry through the x-y plane.
e) reflective symmetry through the y-z plane.
f) reflective symmetry through the z-x plane.

2. Set up the coordinate system, such that the origin is at the center of the parallelogram. x-axis parallels to one pair of sides and y-axis perpendicular to the x-axis. The symmetries are:
a) 180 degree rotational symmetry around z-axis.
b) reflective symmetry through the x-y plane.
c) (double) reflective symmetry through the z-x plane and y-z plane.
3. Set up the coordinate system, such that the origin is at the center of the rectangle. x-axis passes one pair of corners and y-axis passes the other pair of corners of the rhombus. The symmetries are:
a) 180 degree rotational symmetry around z-axis.
b) 180 degree rotational symmetry around x-axis.
c) 180 degree rotational symmetry around y-axis.
d) reflective symmetry through the x-y plane.
e) reflective symmetry through the y-z plane.
f) reflective symmetry through the z-x plane.