First, the definition of congruent graphics
An congruent figure refers to two or more figures, all the corresponding edges are equal, and all the corresponding angles are equal, that is, they can overlap by translation, rotation or flipping. For example, two squares are congruent graphs if their sides are equal.
Second, the judgment of congruent graphics
There are several common methods to judge whether two figures are congruent:
1, SSS: If three sides of two triangles are equal, then the two triangles are congruent.
2. Angular edge (SAS): If two sides and included angles of two triangles are equal respectively, then the two triangles are congruent.
3. Angle and Angle (ASA): If two angles and two sides of two triangles are equal respectively, then the two triangles are congruent.
4. Angular edge (AAS): If two angles of two triangles are equal, then the two triangles are the same.
5. Right-angled side (RHS): If two triangles are right-angled triangles and the hypotenuse is equal to a right-angled side, then the two triangles are congruent.
Third, the nature of congruent graphics
Congruent graphics have the following properties:
1. Corresponding parts are equal: if two graphs are congruent, their corresponding parts (edge, angle, area, perimeter, center, symmetry axis, etc. ) are all equal.
2. Invariance: If a figure is still congruent with the original figure after translation, rotation or flipping, then this transformation is called congruent transformation. The congruent transformation will not change the size and shape of the graph.
3. Transitivity: If A and B are congruent and B and C are congruent, then A and C are congruent.
4. Reflexivity: Any figure is congruent with itself.
Art and Mathematics of Congruent Graphics and Their Transformation
First, the combination of congruent transformations
Any plane figure can be translated, rotated or flipped to get a new figure that is congruent with it. For example, a square can be translated and rotated by 90 to obtain a new square that is congruent with it.
Second, the application of congruent transformation in art
In artistic creation, there is a sense of beauty called symmetry, which means that a pattern or object regularly appears on a certain axis or center. Symmetry can be regarded as a special congruent transformation. For example, the left and right wings of a butterfly are symmetrical about the central axis. Symmetry can increase the sense of stability and harmony of patterns or objects.
Third, the application of congruence transformation in mathematics
There is a branch of mathematics called group theory, which studies some sets with specific structures and operation rules. Among them, an important set is called permutation group, which contains all possible ways to rearrange (replace) a given set. Permutation group can be used to describe the set of new graphics produced by congruent transformation of plane graphics.