Concepts and Connections

Shape

Symmetry--Concepts and Connections

Directed Activity Line of symmetry. The OneLook Dictionary defines symmetry as “an attribute of a shape or relation; exact correspondence of form on opposite sides of a dividing line or plane; or balance among the parts of something.” In geometry, symmetry provides a natural basis for analyzing and classifying objects. For instance, each object in Table 4.1 may be divided into two parts, each a mirror-image of the other, using a vertical line of symmetry. Objects in Table 4.1 have both a vertical and a horizontal line of symmetry. Symmetries based on dividing-lines are called line symmetries.

Table 4.1: Objects Having a Vertical Line of Symmetry

To find a line of symmetry, you need only locate two pair of corresponding points, connect each pair of corresponding points with a line segment, then find the midpoint of each segment. The line of symmetry passes through the midpoints of those segments (Why?). Alternatively, you may connect one pair of corresponding points with a line segment. The line of symmetry is the perpendicular bisector of that segment (Why?). Figure 4.4 illustrates the first of these procedures. Given the approach taken, what can you say concerning the distance of any pair of corresponding points from the line of symmetry?

Figure 4.4: Locating a Line of Symmetry

In Table 4.2, each object has at least two lines of symmetry, one vertical and the other horizontal. Figure 4.5 uses pairs of corresponding points to locate both lines of symmetry.

Table 4.2: Objects Having Vertical and Horizontal Lines of Symmetry

Figure 4.5: Locating Vertical and Horizontal Lines of Symmetry

Directed Activity Point of symmetry. An inspection of the objects in Table 4.3 suggests the existence of a different sort of symmetry, one based on rotations rather than reflections. For instance, each object in Table 4.3 could be rotated through an angle of 180° without changing its appearance (ignoring coloring). Objects having attributes of this sort are said to have turn symmetry. Turn symmetries are characterized by a turn center and a turn angle. Two additional objects with turn symmetry are seen in Table 4.4.

Table 4.3: Objects with Turn Symmetry

To find a turn center, you need only locate two pair of consecutive, corresponding points. Connect each pair of points with a line segment, then find the perpendicular bisector of each segment (Why?). The turn center is located at the intersection of these perpendicular bisectors. The turn angle is determined by two segments, each having one endpoint at the turn center and the other endpoint at two consecutive, corresponding points (See Figures 4.6 and 4.7).

Figure 4.6: Locating the Turn Center

Figure 4.7: Locating the Turn Angle

When searching for lines of symmetry in polygons, it is easy to make false associations with diagonals, altitudes, angle bisectors, perpendicular bisectors and other common interior lines. Table 4.4 illustrates this point. Note that, while the perpendicular bisectors of the sides of the rectangle do coincide with its lines of symmetry, the rectangle’s diagonals do not. On the other hand, both the perpendicular bisectors of the sides and the diagonals of a square do coincide with its lines of symmetry. Under what circumstances will a polygon’s lines of symmetry coincide with its interior lines? Why?

Original Rectangle

Reflected Rectangle

Original Square

Reflected Square

Table 4.4: Lines of Symmetry vs. Interior Lines