Understanding Rotations in Geometry: Center, Angle, and Direction Explained

rotation (or turn)

Rotation, also known as a turn, is a transformation in geometry where an object is turned or pivoted around a fixed point called the center of rotation

Rotation, also known as a turn, is a transformation in geometry where an object is turned or pivoted around a fixed point called the center of rotation. This transformation can be thought of as rotating a shape by a certain angle.

To perform a rotation, we need to know three things: the center of rotation, the angle of rotation, and the direction of rotation (clockwise or counterclockwise).

The center of rotation is a fixed point, typically labeled with a letter, such as point O. This point remains fixed and acts as the pivot or axis around which the shape rotates.

The angle of rotation determines how much to rotate the shape. It is usually measured in degrees or radians. A positive angle indicates a counterclockwise rotation, while a negative angle indicates a clockwise rotation.

The direction of rotation determines whether the shape is rotated in a clockwise or counterclockwise direction. This is important because when describing rotations, the direction is mentioned to avoid ambiguity. Clockwise rotations are depicted by using a negative angle, while counterclockwise rotations are depicted using a positive angle.

When performing a rotation, each point of the shape moves along a circular path around the center of rotation. The distance each point moves is the same, and the resulting shape is congruent (the same size and shape) to the original shape.

To describe a rotation, we often use coordinate notation. If we have a point P(x, y) that is being rotated counterclockwise around the center of rotation O by an angle of θ, the new coordinates of P after rotation can be found using the following formulas:

x’ = (x – h) * cos(θ) – (y – k) * sin(θ) + h
y’ = (x – h) * sin(θ) + (y – k) * cos(θ) + k

In these formulas, (h, k) represent the coordinates of the center of rotation.

It’s important to note that the order of operations matters when applying multiple transformations. If we want to perform multiple rotations, translations, or other transformations, we should apply them in the specific order to get the desired result.

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