REFLECTIONS
• A reflection can be seen in water, in a mirror, or in a
shiny surface.
• An object and its reflection have the same shape
and size, but the figures face in opposite directions.
REFLECTIONS
• In mathematics, the reflection of an object is called
its image.
• If the original object (the preimage) was labeled
with letters, such as polygon ABCD, the image may
be labeled with the same letters followed by a
prime symbol, A'B'C'D'.
REFLECTIONS
• The line (where a mirror may be placed) is called
the line of reflection.
• The distance from a point to the line of reflection is
the same as the distance from the point's image to
the line of reflection.
REFLECTIONS
• A reflection can be thought of as folding or "flipping" an
object over the line of reflection.
Remember:
Reflections are
FLIPS!!!
REFLECTIONS OVER A LINE
• A reflection over a line k
(notation rk) is a
transformation in which
each point of the original
figure (pre-image) has an
image that is the same
distance from the line of
reflection as the original
point but is on the opposite
side of the line.
• Under a reflection, the figure
does not change size.
REFLECTIONS OVER A LINE
• The line of reflection is the
perpendicular bisector of the
segment joining every point and
its image.
• A line reflection creates a figure
that is congruent to the original
figure and is called an isometry (a
transformation that preserves
length).
• Since naming (lettering) the figure
in a reflection requires changing
the order of the letters (such as
from clockwise to
counterclockwise), a reflection is
more specifically called a nondirect or opposite isometry.
REFLECTIONS OVER A LINE
• Properties preserved under a line reflection:
1.
2.
3.
4.
5.
distance (lengths of segments are the same)
angle measures (remain the same)
parallelism (parallel lines remain parallel)
colinearity (points stay on the same lines)
midpoint (midpoints remain the same in each figure)
• Properties NOT preserved
under a line reflection:
1.
orientation (order is reversed)
REFLECTIONS OVER A LINE
• Reflecting over the
x-axis (the x-axis is
the line of
reflection):
• the x-coordinate
remains the same,
but the y-coordinate
is transformed into its
opposite.
REFLECTIONS OVER A LINE
• Reflecting over the
y-axis (the y-axis is
the line of
reflection):
• the y-coordinate
remains the same,
but the x-coordinate
is transformed into its
opposite.
REFLECTIONS OVER A LINE
• Reflecting over the line y = x
or y = -x (the line y = x or y =
-x is the line of reflection):
• When you reflect a point
across the line y = x, the
x-coordinate and the ycoordinate change
places.
• When you reflect a point
across the line y = -x, the
x-coordinate and the ycoordinate change
places and are negated
(the signs are changed).
REFLECTIONS OVER A LINE
• Reflecting over any line:
• Each point of a reflected
image is the same distance
from the line of reflection
as the corresponding point
of the original figure.
• In other words, the line of
reflection lies directly in the
middle between the figure
and its image -- it is the
perpendicular bisector of
the segment joining any
point to its image.
• Keep this idea in mind
when working with lines of
reflections that are neither
the x-axis nor the y-axis.
REFLECTIONS OVER A POINT
• 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.
The diagram above shows
points A and C reflected
through point P. Notice that
P is the midpoint of segment
AA’ and segment CC’.
REFLECTIONS OVER A POINT
• 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.
REFLECTIONS OVER A POINT
• Properties preserved under a point reflection:
1.
2.
3.
4.
5.
6.
distance (lengths of segments are the same)
angle measures (remain the same)
parallelism (parallel lines
remain parallel)
colinearity (points stay on the same lines)
midpoint (midpoints remain the same in each figure)
orientation (lettering order remains the same)
REFLECTIONS OVER A POINT
• 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 half-turn (or a rotation of 180º).
• If the point of reflection is P, the notation may be
expressed as a rotation RP, 180°
or simply RP.
REFLECTIONS OVER A POINT
• 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.
• When working with the graph of y = f(x), replace x
with -x and y with -y.
REFLECTIONS OVER A POINT
• 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.
REFLECTIONS OVER A POINT
• 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.