The diagram above
shows points A
and C reflected through point P. Notice that
P is the midpoint of segments
. 
A
point reflection
exists
when a figure is built around a single point called the center of the
figure, or point of reflection. For every point in the figure, there is another point
found directly opposite it on the other side of the center such that the
point of reflection becomes the midpoint of the segment joining
the point with its image. Under a point reflection,
figures do not change size.

A point
reflection creates a figure that is congruent to the
original figure and is called an isometry
(a transformation that preserves length). Since the orientation in a point reflection
remains the same
(such as counterclockwise seen in this diagram), a point reflection is more
specifically called a direct isometry. 

Properties preserved (invariant) under a
point
reflection:
1. distance (lengths of segments are the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. colinearity (points stay on the same lines)
5. midpoint (midpoints remain the same in each figure)
6. orientation (lettering order remains the same) 

Definition: A reflection
in a point P is a transformation of the plane such that
the image of the fixed point P is P and for all other
points, the image of A is A' where P is the midpoint
of .
A figure
that possesses point symmetry can be recognized because it will be
the same when rotated 180 degrees. In this same manner, a
point reflection can also be called a halfturn (or a rotation of
180º). If the point of reflection is P, the notation
may be expressed as a rotation
or simply
. 

Point Reflection in the Coordinate Plane:
While any point in the
coordinate plane may be used as a point of reflection, the most
commonly used point is the origin. Assume that the origin is
the point of reflection unless told otherwise. You may see various
notations such as
(rotation statements)
or where point O
is the origin
or (reflection statement)
.

Triangle
A'B'C' is the image of triangle ABC after a
point reflection in the
origin.
Imagine
a straight line connecting A to A' where the origin is the
midpoint of the segment. 


Notice how
the coordinates of triangle A'B'C' are the same
coordinates as triangle ABC, BUT the signs have
been changed.
Triangle
ABC has been reflected in the origin. 

