subject: Reflection Of A Point In A Line [print this page] In geometric, the determinations of reflection of a point in a line translation is performed only through a given point or line. In reflection of a point in a line, the basic ideas of reflection are the transformation of a point to a reflected point that is the equal length of the opposite side of a line. The transformation of reflections in two right angular axes produce a rotation of straight angle (180), that is a half turn.
Discussion on Reflection of a Point in a Line
The reflection of a point in a line let us consider the point is P and line is L. Then the reflection of point P is P. Here, PP is a vertical line to line L. Let us take M is the mid point of PP. Therefore, PM = MP and L is labelled axis of symmetry or axis of reflection.
The reflection flips the geometric point or object or shape or image across a line L. The new mirrored object of a point is a reflected object of the original geometric object of a point.
Reflection construction follow the given procedures.
Step 1: To determine the distance from the given object of a point to the required line L or mirror line.
Step 2: To plot the reflected point on the opposite side of line L. The reflected point P' from the equal distance of
a line L or mirror line.
Step 3: To get the reflected object of point and form the reflected object or image or shape.
For example,the transformation of three geometric points are P(x, y). It is transformed by using the reflection of a point in a line, that is the line L or mirror line or line of reflection.
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Reflection of a Point in a Line X - Axis and Y - Axis:
Reflection of a point in a line to describe the X-axis is consider the mirror line or axis of reflection. Therefore, we change the given geometric object point into x = x and y = -y.
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Reflection of a point in a line to describe the Y-axis is consider the mirror line or axis of reflection.. Therefore, we change the given geometric object point into x = -x and y = y.
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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)
6. orientation (lettering order NOT preserved. Order is reversed.)
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.
