Transformations Unit, Lesson 1 Reflections Transformations Unit, Lesson 1

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.

Remember: Reflections are FLIPS!!! 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 non-direct or opposite isometry.

Reflections over a line Properties preserved under a line reflection: 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: 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 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).

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: 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.