# Reflections

A reflection is one of the three kinds of transformations of **plane** figures which move the figures but do not change their shape. It is called a reflection because figures after a reflection are the mirror images of the original ones. The reflection takes place across a line called the "line of reflection." Figure 1 shows a triangle ABC and its image A'B'C'. Each individual point and its image lie on a line which is **perpendicular** to the line of reflection and are equidistant from it. An easy way to find the image of a set of points is to fold the **paper** along the line of reflection. Then, with the paper folded, prick each point with a pin. When the paper is unfolded the pin pricks show the location of the images.

One reflection can be followed by another. The position of the final image depends upon the position of the two lines of reflection and upon which reflection takes place first.

If the lines of reflection are **parallel**, the effect is to slide the figure in a direction which is perpendicular to the two lines of reflection, and to leave the figure "right side up." This combined **motion**, which does not rotate the figure at all, is a "translation." The distance the figure is translated is twice the distance between the two lines of reflection and in the first-line to second-line direction.

If the lines of reflection are not parallel, the effect will be to rotate the figure around the point where the two lines of reflection cross. The **angle** of **rotation** will be twice the angle between the two lines and will be in a first-line to second-line direction. Because a figure can be moved anywhere in the plane by a combination of a translation and a rotation and can be turned over, if necessary, by a reflection, the combination of four or five reflections will place a figure anywhere on the plane that one might wish.

Someone who, instead of lifting a heavy slab of stone, moves it by turning it over and over uses this idea. In moving the stone, however, one is limited to the lines of reflection that the edges of the stone provide. Some last adjustment in the slab's position is usually required.

Reflections can also be accomplished algebraically. If a point is described by its coordinates on a **Cartesian coordinate plane**, then one can write equations which will connect a point (x, y) with its reflected image (x', y'). Such equations will depend upon which line is used as the line of reflection. By far the easiest lines to use for this purpose are the x-axis, the y-axis, the line x = y, and the line x = -y. Figures 4 and 5 show two such reflections. In Figure 4 the line of reflection is the y-axis. As the figure shows, the y-coordinates stay the same, but the x-coordinates are opposites: x' = -x and y' = y. One can use these equations in two ways. If a point such as (4,7) is given, then its image, (-4,7), can be figured
out by substituting in the formulas. If a set of points is described by an equation such as 3x-2y = 5, then the equation of the image, -3x'-2y' = 5, can be found, again by substitution.

When the line of reflection is the line x = y, as in Figure 5, the equations for the reflection will be x = y', and y = x'. These can be used the same way as before. The image of (3,1) is (1,3), and the image of the **ellipse** x^{2} + 4y^{2} = 10 is 4x^{2} + y^{2} = 10 (after dropping the
primes). The effect of the reflection was to change the major axis of the ellipse from horizontal to vertical.

When the line of reflection is the x-axis, the y-coordinates will be equal, but the x-coordinates will be opposites: x' = -x and y' = y.

When the line of reflection is the line x = -y, these equations will effect the reflection: x' = -y and y' = -x.

The idea behind a reflection can be used in many ways. One such use is to test a figure for reflective **symmetry**, to test whether or not there is a line of reflection, called the "axis of symmetry" which transforms the figure into itself. Letters, for example, are in some instances symmetrical with respect to a line and sometimes not. The letters A, M, and W have a vertical axis of symmetry; the letters B, C, and E, a horizontal axis; and the letters H, I, and O, both. (This symmetry is highly dependent on the typeface. Only the plainest styles are truly symmetrical.) If there is an axis of symmetry, a mirror held upright along the axis will reveal it.

While recognizing the reflective symmetries of letters may not be of great importance, there are situations where reflection is useful. A building whose facade has reflective symmetry has a pleasing "balance" about it. A reflecting pool enhances the scene of which it is a part. Or, contrarily, artists are admonished to avoid too much symmetry because too much can make a picture dull.

When tested analytically, a figure will show symmetry if its equation after the reflection is, except for the primes, the same as before. The **parabola** y = x^{2} is symmetrical with respect to the y-axis because its transformed equation, after dropping the primes, is still y = x^{2}. It is not symmetrical with respect to the x-axis because a reflection in that axis yields y = -x^{2}. Knowing which axes of symmetry a graph has, if any, is a real aid in drawing the graph.

## Resources

### Books

Kazarinoff, Nicholas D. *Geometric Inequalities.* Washington, DC: The Mathematical Association of America, 1961.

Pettofrezzo, Anthony. *Matrices and Transformations.* New York: Dover Publications, 1966.

Yaglom, I.M. *Geometric Transformations.* Washington, DC: The Mathematical Association of America, 1962.

J. Paul Moulton

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