Rotating 180 Degrees About a Given Center

A half turn spins a figure 180 degrees, so every point travels exactly halfway around a circle centered at the rotation point. Half of a full circle is a straight line through the center, so the point, the center, and the image all line up. The image lands on the opposite ray from the point, at the same distance from the center. That one fact, straight line plus equal distance, is all a 180 degree rotation is.

Because the point, center, and image are collinear with the center exactly between them, the center is the midpoint of the segment joining a point to its image. That gives a shortcut with coordinates: if the center is (h, k) and the original point is (x, y), the image is (2h - x, 2k - y). You reach it by going from x to h and continuing the same distance past it, and the same for y. Run that rule backward and you get the other practice type: given a point and its image, the center is just their midpoint, ((x1+x2)/2, (y1+y2)/2).

This topic sits next to reflections and 90 degree rotations in the transformations unit, and it is really the simplest case of rotation because there is no need to track an angle of turn, only a center. It also connects to point symmetry: a figure that maps onto itself under a 180 degree turn about some center, like a parallelogram about the intersection of its diagonals, is called centrally symmetric. Seeing 180 degree rotation as 'reflect through a point' rather than 'turn by an angle' makes later rotation problems, at 90 or 270 degrees, easier to picture.