Chapter 9 Congruence, Symmetry and Similarity Section 9.4 Symmetry

In geometry symmetry has to do with how “balanced” or “even” an object looks. Many objects both man made and naturally occurring exhibit different kinds of symmetry. A plane figure has symmetry if you can trace it on a piece of paper move the paper to a different position and have it match up with the original figure exactly. Like many other subjects in the study of mathematics there is a more precise way to describe this concept. butterflysnowflake Britney SpearsF-22 Raptor Line (Reflection) Symmetry A shape has a line (reflection) symmetry if there is a line about which you can reflect the shape so that the result is the original shape. The line that a shape can be reflected across is called a line of symmetry of the shape. Lines of symmetry are red dashed lines in the shapes below.

Representing Line (Reflection) Symmetries The way the line symmetries are often represented on polygons is by labeling the vertices with a letter (or a number) and showing what vertices are interchanged by the reflection through the line of symmetry. Here are some examples: A B C D A B C D Reflection through a vertical line of symmetry Reflection through a horizontal line of symmetry A B C D A B C D Reflection through diagonal

Rotational (Turn) Symmetries Rotational (Turn) symmetries rotates the object about a fixed (center) point so that it turns back onto itself. Here are some examples below. Rotational symmetries are represented in an analogous manner that reflection symmetries are represented. Each vertex is labeled with a letter (or number) and where that vertex rotated to is shown. A B C D A B C D A B C D A B C D 90  rotation 180  rotation 270  rotation 360  rotation

Each of the triangles below is a symmetry transformation applied to the equilateral triangle to the right. Name the symmetry transformation that has been applied. All rotations are clockwise. How do you figure out the number of degrees in the rotation? A B C C B A B C A B A C A B C C A B A C B Reflection of line of symmetry through vertex B 240  rotation Reflection of line of symmetry through vertex C 360  rotation 120  rotation Reflection of line of symmetry through vertex A

Symmetries of Regular Polygons The number of symmetries of a regular polygon with n sides is 2n. There are n reflection symmetries and n rotational symmetries. For example a regular pentagon has 5 reflection symmetries and 5 rotational symmetries. To get the degrees of each rotational symmetry take 360  5 = 72 . Start with 72  and keep adding 72  until you get to 360 . 72  rotation 144  rotation 72  +72  =144  216  rotation 144  +72  =216  288  rotation 216  +72  =288  360  rotation 288  +72  =360 