Introduction A line of symmetry,, is a line separating a figure into two halves that are mirror images. Line symmetry exists for a figure if for every point P on one side of the line, there is a corresponding point Q where is the perpendicular bisector of. 1
Introduction, continued From the diagram, we see that is perpendicular to. The tick marks on the segment from P to R and from R to Q show us that the lengths are equal; therefore, R is the point that is halfway between. Depending on the characteristics of a figure, a figure may contain many lines of symmetry or none at all. In this lesson, we will discuss the rotations and reflections that can be applied to squares, rectangles, parallelograms, trapezoids, and other regular polygons that carry the figure onto itself. Regular polygons are two-dimensional figures with all sides and all angles congruent. 2
Introduction, continued Squares Because squares have four equal sides and four equal angles, squares have four lines of symmetry. If we rotate a square about its center 90, we find that though the points have moved, the square is still covering the same space. Similarly, we can rotate a square 180, 270, or any other multiple of 90 with the same result. 3
Introduction, continued We can also reflect the square through any of the four lines of symmetry and the image will project onto its preimage. Rectangles A rectangle has two lines of symmetry: one vertical and one horizontal. Unlike a square, a rectangle does not have diagonal lines of symmetry. If a rectangle is rotated 90, will the image be projected onto its preimage? What if it is rotated 180? 4
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Introduction, continued If a rectangle is reflected through its horizontal or vertical lines of symmetry, the image is projected onto its preimage. A, B' Horizontal reflection B, A' Vertical reflection A, D' B, C' D, C' C, D' D, A' C, B' 6
Introduction, continued Trapezoids A trapezoid has one line of symmetry bisecting, or cutting, the parallel sides in half if and only if the nonparallel sides are of equal length (called an isosceles trapezoid). We can reflect the isosceles trapezoid shown below through the line of symmetry; doing so projects the image onto its preimage. However, notice that in the last trapezoid shown on the next slide, is longer than, so there is no symmetry. The only rotation that will carry a trapezoid that is not isosceles onto itself is 360. 7
Introduction, continued 8
Introduction, continued Parallelograms There are no lines of symmetry in a parallelogram if a 90 angle is not present in the figure. Therefore, there is no reflection that will carry a parallelogram onto itself. However, what if it is rotated 180? 9
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Key Concepts Figures can be reflected through lines of symmetry onto themselves. Lines of symmetry determine the amount of rotation required to carry them onto themselves. Not all figures are symmetrical. Regular polygons have sides of equal length and angles of equal measure. There are n number of lines of symmetry for a number of sides, n, in a regular polygon. 11
Common Errors/Misconceptions showing a line of symmetry in a parallelogram or rhombus where there isn t one missing a line of symmetry 12
Guided Practice Example 1 Given a regular pentagon ABCDE, draw the lines of symmetry. 13
Guided Practice: Example 1, continued 1. First, draw the pentagon and label the vertices. Note the line of symmetry from A to. 14
Guided Practice: Example 1, continued 2. Now move to the next vertex, B, and extend a line to the midpoint of. 15
Guided Practice: Example 1, continued 3. Continue around to each vertex, extending a line from the vertex to the midpoint of the opposing line segment. Note that a regular pentagon has five sides, five vertices, and five lines of reflection. 16
Guided Practice: Example 1, continued 17
Guided Practice Example 3 Given the quadrilateral ABCE, the square ABCD, and the information that F is the same distance from A and C, show that ABCE is symmetrical along. 18
Guided Practice: Example 3, continued 1. Recall the definition of line symmetry. Line symmetry exists for a figure if for every point on one side of the line of symmetry, there is a corresponding point the same distance from the line. We are given that ABCD is square, so we know @. We also know that. is symmetrical along We know @. 19
Guided Practice: Example 3, continued 2. Since @ and @, is a line of symmetry for where. 20
Guided Practice: Example 3, continued 3. has the same area as because they share a base and have equal height. @, so. 21
Guided Practice: Example 3, continued 4. We now know is a line of symmetry for and is a line of symmetry for, so and quadrilateral ABCE is symmetrical along. 22
Guided Practice: Example 3, continued 23