For AI agents: use /llms.txt for the Nakafa content index.
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 .
The center of reflection is the midpoint of the joining segment. The average of each pair of coordinates therefore equals the corresponding center coordinate:
A point at the center remains fixed. For example, gives .
Determine the image of a half-turn for the point .
This means we are reflecting point over the center point .
Here, , , , and .
Calculate twice each center coordinate, then subtract the original point coordinate:
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 original equation of line :
Replacing and back to and , the equation of the image line is:
Alternatively, it can be written as .
The problems use a half-turn about a given center. The third reflects a line, so its equation is transformed.
For a point, check that the center is the midpoint of the original and image. For a line, express the original coordinates in terms of the image coordinates before substituting.
Solution 1
Center , point . So .
Thus, the image of point is .
Solution 2
Center , point . So .
Thus, the image of point is .
Solution 3
Center . Line .
Substitute into the line equation:
Image line equation: or .
Published: . Updated: .
