Symmetry

Definition : A line of symmetry divides a set of points into two halves, each being a reflection of the other. Each image point is also a point of the set.

Definition : A set of points has line symmetry if a line can be drawn through it so that every image point on one side of the line is also a point of the set.
One way to test if a figure has line symmetry is to fold it in half along the supposed line of symmetry and see if the two sides are mirror images of each other. The following have one or more lines of symmetry. The number of symmetry lines is labeled.

Notice that while polygons have a limited number of symmetry lines, a circle has an infinite amount. Any line that passes through the center of the circle is a line of symmetry. The following do not have lines of symmetry.

A line of symmetry through a line segment is its perpendicular bisector.

Similarly, a line of symmetry of an angle is the line which bisects the angle.

Regular polygons have the same number of symmetry lines as their sides.

Definition : A reflection point is the midpoint of a line segment joining any point to its image point.

A set of points has point symmetry if there exists a point P, and every reflection of a point of the set through point P coincides with another point of the set.

In point symmetry, there is one point through which every other point is reflected. That point is also its own reflection. One way to test if a figure has point symmetry is to turn it upside-down and see if it looks the same. To understand the difference between line symmetry and point symmetry, compare the following examples:

A set of points has rotational symmetry, if it can be rotated with reference to a fixed point with a rotation less than 360; so that it coincides or overlaps with the same set of points.

Point symmetry is just one type of rotational symmetry where the figure coincides with itself when rotated 180. The following are examples of rotational symmetry where the figure is rotated at angles other than 180. They have rotational symmetry, but not point symmetry.