12.2.1 Symmetry
Key Objectives • Identify and describe symmetry in geometric figures. Key Terms • A figure has symmetry if there is a transformation of the figure such that the image coincides with the preimage. • A figure has line symmetry (or reflection symmetry) if it can be reflected across a line so that the image coincides with the preimage. • The line of symmetry (also called the axis of symmetry) divides the figure into two congruent halves. • A figure has rotational symmetry (or radial symmetry) if it can be rotated about a point by an angle greater than 0° and less than 360° so that the image coincides with the preimage. • A three-dimensional figure has plane symmetry if a plane can divide the figure into two congruent reflected halves. • A three-dimensional figure has symmetry about an axis if there is a line about which the figure can be rotated by an angle greater than 0° and less than 360° so that the image coincides with the preimage. Example 1 Identifying Line Symmetry
Whether a figure has line symmetry is determined in this example. This pentagon has one line of symmetry. Notice that the portions of the object on either side of the line are reflections of each other. Or, if the object were reflected across the line of symmetry, the image would lie exactly on top of the preimage. To verify that this is the only line of symmetry for this object, try positioning the line at other locations through the pentagon and determine whether it produces reflections. Whether a figure has line symmetry is determined in this example. This figure does not have a line of symmetry. There is no line across which the object could be reflected so that the image would lie exactly on top of the preimage.
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