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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. Remember that a
reflection is a flip.
Under a reflection, the figure does not change size.
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 non-direct or opposite
isometry.
Properties preserved (invariant) under a line
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)
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6. orientation
(lettering order NOT
preserved. Order is reversed.) |
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Definition: A reflection
is an isometry where if l is any line and P is any point
not on l, then rl(P) = P' where
l is the perpendicular bisector of
and if
 then rl(P)
= P.
So what is this definition saying:
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Part I (up to the word "and"):
Here we see line l and point P not
on line l. The reflection of point P
in this line will be point P'. This is
stated by rl(P)=P'.
The line l will be the perpendicular bisector
of the segment joining point P to point
P'.
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Part II (after the word "and"):
The second part of the definition deals with
point P being on line l.
says P is an element of line l .
If P in on the line, then it is its own
reflection in line l. This is
stated by rl(P) = P.
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Reflections in the Coordinate Plane:
Reflecting over the
x-axis:
(the x-axis as the
line of reflection) |
When you reflect a point across the
x-axis, the
x-coordinate remains the same,
but the y-coordinate is transformed into
its opposite.
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Hint:
If you forget the rules for reflections when graphing, simply fold your graph paper along the
line of reflection (in this example the
x-axis) to see
where your new figure will be located. Or you can measure how
far your points are away from the line of reflection to locate your new
image. Such processes will allow you to see what is happening to the
coordinates and help you remember the rule. |
Reflecting
over the y-axis:
(the y-axis as the
line of reflection) |
When you reflect a point across the
y-axis, the
y-coordinate remains the same,
but the x-coordinate is transformed into
its opposite.
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The
reflection of the point
(x, y)
across the
y-axis is the point
(-x, y).
or
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Reflecting
over the line y = x or
y = -x:
(the lines y =
x or y = -x as the lines of reflection) |
When you reflect a point across the line
y = x, the
x-coordinate and the y-coordinate change places. When you reflect
a point across the line y = -x, the x-coordinate and the
y-coordinate
change places and are negated (the signs are changed).
| 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.
Notice
how each point of the original figure and its image are the same distance away
from the line of reflection (which can be easily counted in this diagram since
the line of reflection is vertical).
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