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Rotation and its Angle
In plane geometry, rotation means to turn the whole plane by a given angle, around a fixed point called the "center of rotation."
Move around the red points A_{1}, A_{2}, A_{3}, A_{4}, or the green point V on the unit circle.
You can see how this changes the positions of the blue points B_{1}, B_{2}, B_{3}, and B_{4}. They are the images of points A_{1}, A_{2}, A_{3}, and A_{4} under a counterclockwise rotation by the angle UOV, with vertex at the origin O.
Rotations shown in the interactive worksheet above are based on the point (0,0), the origin of the XYplane, as their center. The angle of rotation is controlled by point P (shown in green) on the unit circle.
As you move point P along the unit circle, you see the corresponding angle value, in degrees. At the same time, each red point is rotated around the origin (0,0), by the same angle, from the position of the corresponding blue point. Point A2 is the rotated image of point A1, and so on.
Rotations are examples of geometrical transformations called isometries, because they not only preserve the shapes of geometric figures but also their size.
Rotations are also examples of Linear Transformations, a much more general type of transformations studied in Linear Algebra.
Linear transformations are often expressed using matrices (rectangular arrays of rows and columns filled with numbers).
The formula below shows how to define a rotation of the XYplane around the origin (0,0) by a given angle (theta) using its associated 2x2 matrix.
In relation to the formula below, ( cos(theta), sin(theta) ) are the coordinates of point P (shown in green) on the unit circle,
while Q = (x,y) represents any of the blue points A1,..,D1, and R represents the corresponding red, rotated point A2,..,D2.
In the interactive GeoGebra worksheet applet presented above, you can move the blue points around the screen, not only one at a time but also two at a time by segments, meaning, you can move each one of A_{1}B_{1}, B_{1}C_{1}, C_{1}D_{1}, and D_{1}A_{1}, the sides of the blue polygon.

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