Reflection Across a Line - Online Math Tutor Juan Castaneda, MS



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Reflection Across a Line
The Axis of Reflection is the line AB shown in green in the applet below

Move around any red point, or the green points A and B, to see the reflection effect on the blue points:

Each blue point is the reflected, mirror image of a red point across the line AB.
The blue points automatically react to the movements of their corresponding red point, and/or to those of the axis of reflection (the green line).

In plane geometry, Reflection across a line is defined by moving each point perpendicularly across a straight line (known as the Axis of Reflection), as far into the other side of the given axis, as the distance from the original point to the axis of reflection.

For example, when the axis of reflection is vertical, points originally to its right move onto the left-hand side while points originally to its left move onto the right-hand side. On the other hand, when the axis of reflection is horizontal, points originally above it move onto the lower side while points originally below it move onto the upper side.

In the interactive Geogebra worksheet above, by individually moving the green points U and/or V, you can move the Axis of Reflection -green solid line- into any position (vertical, horizontal, or diagonally slanted). This allows you to see the effects of the reflection in each case, because the red shape (given by points A2,B2,C2,D2) is always presented as the reflected image of the blue shape (given by points A1,B1,C1,D1), across the axis of reflexion UV.

By moving the blue points, you will soon notice this property of reflections: the farther apart from the axis the original points are, the farther apart from the axis their reflected images will be but on the other side of the line. And vice-versa, the closer the original points are to the axis of reflection, the closer their reflected images will be to the same axis, just on the other side of it.

Another property of reflections is that the axis of reflection itself remains fixed under the reflection it defines. You will notice this not by moving the axis UV but by leaving it fixed, while bringing onto it any one of the blue points (A1,B1,C1,D1). When you place a blue point exactly on the green line (the axis of reflection) you will see the corresponding red point (A2,B2,C2,D2) simultaneously moving onto the very same spot.

You may also notice an interesting property that somehow connects reflections with Rotation. When you move around one of the green points (either U or V), the axis of reflection rotates around the other point, the one that remains fixed. At the same time the red shape, being the reflected image of the blue shape across the green line, automatically rotates around the same fixed point (U or V) the axis of reflection is rotating around.

All reflections are examples of geometrical transformations called isometries, because they not only preserve the shapes of geometric figures but also their size.

Some reflections are also Linear Transformations. Not all reflections are linear transformations but only those where the axis of reflection goes through the origin (0,0) of the XY-plane. This is because Linear Transformations need to leave the origin fixed. Any geometric transformation of the XY-plane that moves the origin (0,0) to somewhere else, is not a linear transformation. However, when the axis of reflection goes through the origin (0,0) of the XY-plane, that particular reflection will be a linear transformation.

You may want to try this little activity:

  1. Put the Axis of Reflection in a vertical position along the Y-axis (right on top of it). For example, you can place point U at coordinates (0, -4), and point V at coordinates (0,6)
  2. Place blue point A1 at coordinates (3, -2), and blue point B1 at coordinates (3,3)
  3. Once you do this, red point A2 should automatically relocate to (-3, -2), and red point B2 sholud place itself at (-3,3)
  4. Now take the blue point C1, and place it right on top of red point B2. You will notice red point C2 will automatically react by placing itself on top of blue point B1
  5. Finally, take the blue point D1, and place it on top of red point A2. At the same time, red point D2 will locate itself right on top of blue point A1.
  6. Notice how now the blue polygon, and the red polygon, both form one and the same rectangle, a rectangle that is symmetric with respect to the Y-axis (the axis of reflection in this particular case)
This activity illustrates another property of reflections across a line. That is, when a polygon is symmetric with respect to the axis of reflection, its reflected image ends up being the same polygon (with a different ordering of the vertices but still the same subset of the XY-plane).

In the interactive GeoGebra worksheet applet presented above, you can move the blue points around the screen, not only one at a time but also two at a time by segments, meaning, you can move each one of A1B1, B1C1, C1D1, and D1A1, the sides of the blue polygon.




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Last review: April 7, 2017