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Geometry Transformations

Rotational Symmetry

A figure has rotational symmetry if it can be rotated around a point so that resulting figures match the original figure. The point around which you rotate is called the center of rotation, and the smallest angle you need to turn is called the angle of rotation.

The number of positions a figure can be rotated to, without bringing in any changes to the way it looks originally, is called its Order of Rotational Symmetry.

 This card has rotational symmetry of order 2. Turn it upside-down and it still looks the same. This shape has rotational symmetry of order 4. Four times, as it completes a turn, it will look the same.

A special type of rotational symmetry is point symmetry. A figure has point symmetry if it can be rotated 180º around a point to match the original figure. It looks the same upside down as right side up.

How many degrees of rotational symmetry have the following polygons?

 Equilateral triangle The order of rotational symmetry of an equilateral triangle is three. The angle of rotation is 120º. Square The order of rotational symmetry of a square is four. The angle of rotation is 90º. Regular pentagon The order of rotational symmetry of a regular pentagon is 5. The angle of rotation is 72º. Regular pentagon The order of rotational symmetry of a regular polygon equals the number of its sides. The angle of rotation is degrees (n=number of sides)

Following are facts about rotational symmetry:

• All figures have at least one order of rotational symmetry. (Rotating a figure 360º will always match the original figure).

Draw a figure with 45º of rotational symmetry.

Regular polygons have a degree of rotational symmetry equal to 360 divided by the number of sides. We can solve $\frac{360}{n}=45$ to find the number of sides of our regular polygon.

$\frac{360}{n}=45\;\Rightarrow\;\frac{360}{45}=n\;\Rightarrow\;n=8$