In a plane, a figure rotates a certain angle around a fixed point to get the change of another figure, which is called rotation. This fixed point is called the rotation center, and the rotation angle is called the rotation angle. If point A on the graph rotates to point A', then these two points are called corresponding rotation points.

Three elements

① center of rotation;

② direction of rotation;

③ Rotation angle.

Main attributes

① The distance from the corresponding point to the center of rotation is equal.

(2) The included angle of the connecting line between the corresponding point and the rotation center is equal to the rotation angle.

③ Graphic consistency before and after rotation.

④ The center of rotation is the only fixed point.

⑤ The included angle of a group of corresponding points is equal to the rotation angle. (Junior high school cannot be directly used as a conclusion)

Rotation definition

In a plane, turning a figure around a figure by an angle in a certain direction is called the rotation of the figure. This fixed point is called the center of rotation and the rotation angle is called the rotation angle.

Rotational characteristics

The rotation of a graph means that each point on the graph moves around a fixed point on the plane at a fixed angle, where: ① the distance from the corresponding point to the rotation center is equal; ② The length of the corresponding line segment is equal to the size of the corresponding angle; ③ The size and shape of the figure have not changed before and after rotation.

Centrally symmetric figures and centrally symmetric figures

Centrally symmetric figure: If a figure can overlap itself after rotating 180 degrees around a certain point, then we say that this figure has formed a centrosymmetric figure.

Central symmetry: If one graph can overlap another graph after rotating 180 degrees around a certain point, then we say that these two graphs form central symmetry. 4. The essence of central symmetry:

On the congruence of two graphs with central symmetry.

For two graphs with central symmetry, the straight lines connecting the symmetrical points pass through and are equally divided by the symmetrical center. For two figures with symmetrical centers, the corresponding line segments are parallel (or on the same straight line) and equal.

Symmetric transformation of points

(1), the point is symmetrical about the origin.

When two points are symmetrical about the origin, the signs of their coordinates are opposite, that is, the symmetrical point of point P(x, y) about the origin is P'(-x, -y) (2), and the point is symmetrical about X.

When two points are axisymmetrical about X, in their coordinates, X is equal and the sign of Y is opposite, that is, the point where point P(x, y) is axisymmetrical about X is P'(X, -y).

(3) The characteristics of the point symmetrical about the Y axis

When two points are symmetrical about Y, Y is equal, and the sign of X is opposite in its coordinates, that is, the point where P(x, y) is symmetrical about Y is P'(-x, y).

(4) Symmetry of straight line y=x

When two points are symmetrical about the straight line y=x, the abscissa and ordinate are reversed, that is, the symmetrical point of P(x, y) about the straight line y=x is P'(y, x).

(5) When two points are symmetrical about the straight line y=-x, the abscissa and ordinate are completely opposite to before, that is, the symmetrical point of P(x, y) about the straight line y=x is P'(-y, -x).

Note: A straight line with y=x is the bisector of an angle passing through one or three quadrants, and a straight line with y=-x is the bisector of an angle passing through two or four quadrants.