Rotation refers to the transformation of a geometric figure around a fixed point, known as the center of rotation, by a specified angle in a given direction. This concept is fundamental in understanding how shapes can be manipulated and repositioned in a coordinate plane, impacting their orientation while maintaining their size and shape.
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When a figure is rotated 90 degrees counterclockwise, the coordinates of each point change according to specific rules: if a point (x, y) is rotated around the origin, it becomes (-y, x).
Rotations can be either clockwise or counterclockwise, and the direction must be specified when describing the transformation.
The distance from any point on the figure to the center of rotation remains constant after rotation, meaning the shape retains its original proportions.
Rotating a figure multiple times can lead to different orientations, but if rotated a full 360 degrees, the figure will return to its original position.
In composition of transformations, rotations can be combined with other transformations such as translations and reflections to achieve complex movements of figures.
Review Questions
How does the center of rotation affect the position of a geometric figure when it undergoes rotation?
The center of rotation plays a crucial role in determining how a geometric figure is repositioned during rotation. If the center is located inside the figure, the points will move along arcs toward their new positions while remaining equidistant from this center. Conversely, if the center is outside the figure, the entire shape will swing around it in a circular path, leading to different orientations without changing the figure's dimensions.
Explain how to calculate the new coordinates of a point after performing a 180-degree rotation about the origin.
To calculate the new coordinates after a 180-degree rotation about the origin, you simply take the original coordinates (x, y) and apply the transformation by changing both signs: the new coordinates will be (-x, -y). This means every point on the figure flips across both axes, resulting in an orientation that is directly opposite to its original position.
Evaluate how combining rotations with other transformations can impact the final orientation and position of a geometric figure.
Combining rotations with other transformations like translations and reflections can lead to complex changes in both orientation and position of a geometric figure. For example, if you rotate a triangle 90 degrees and then translate it 3 units right, its final position will be determined by both operations. Similarly, reflecting it after rotation may result in an entirely different orientation compared to performing these operations in reverse order. Understanding these combinations allows for more advanced manipulation of figures in geometry.
Related terms
Center of Rotation: The fixed point around which a figure is rotated.
The degree to which a figure is rotated about the center of rotation, typically measured in degrees or radians.
Transformation: A general term that encompasses various operations that alter the position or size of a geometric figure, including rotations, translations, and reflections.