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Reflection over a point, often called a half-turn rotation (), is a geometric transformation where each point on an object is mapped to a new position such that the center of reflection becomes the midpoint between the original point and its image.
Suppose the center of reflection is . If a point is reflected over point , its image will lie on the line passing through and , with as the midpoint of the segment .
If a point is reflected over the point , its image's coordinates, , are determined by the formula:
Alternatively, it can be written as:
This means the -coordinate of the image is twice the -coordinate of the center minus the original -coordinate, and the same applies to the -coordinate.
Determine the image of a half-turn for the point .
This means we are reflecting point over the center point .
Here, , , , and .
Using the formula:
Thus, the image of point is .
Determine the image of a half-turn for the line with the equation .
Take an arbitrary point on line . Its image, , after reflection over point is:
Substitute and into the equation of line :
Replacing and back to and , the equation of the image line is:
Alternatively, it can be written as .
Center , point . So .
Thus, the image of point is .
Center , point . So .
Center . Line .
Thus, the image of point is .
Substitute into the line equation:
Image line equation: or .
