For AI agents: use /llms.txt for the Nakafa content index.
The center remains fixed. For a point , rotation preserves and turns the direction from the center through the directed angle . Measure from ray toward ray , taking counterclockwise as positive.
Angles differing by a full turn give the same image. For example, a counterclockwise turn gives the same result as a clockwise turn:
For rotation about the origin, the sine and cosine of the turn angle determine how the coordinates combine. Substituting special-angle values gives shorter rules:
| Angle | Image | Direction |
|---|---|---|
| Quarter turn counterclockwise | ||
| Quarter turn clockwise | ||
| Half-turn |
A point is rotated about the origin by . Determine its image.
Here, , , and .
We know and .
Use a positive angle for this counterclockwise turn:
The image of point is .
Determine the image of the line rotated about the origin by .
Let be any point on the line . Its image, , after a rotation about the origin is:
From this, we get and .
Substitute and into the original line equation :
Replacing and back to and , the equation of the image line is or .
The first two problems use a positive quarter turn. The last uses a half-turn, which reverses both coordinate signs. For a line, substitute into the original equation to map every point on it.
Use the sine and cosine values for the given angle. For a line, express the original coordinates in terms of the image coordinates before substituting into the equation.
Solution 1
Point , . .
Thus, its image is .
Solution 2
Line , .
So and .
Substitute into the line equation: .
Image line equation: or .
Solution 3
Point , . .
Thus, the image of point is .
Published: . Updated: .
