# Central symmetry

Central symmetry is the situation in which there are homologous points with respect to the point that is called the center of symmetry.

In symmetry, to explain it in another way, each point corresponds to another that is at the same distance from the point of symmetry.

To define it formally, the central symmetry can be defined as the product of the fulfillment of the following rule: If we have the points X and X ', both are symmetric with respect to a center (C), if the segment CX is equal to the segment CX' (they are the same length), so X and X 'are equidistant from C.

It should be mentioned that the central symmetry can not only be observed in two segments, but also in polygons, for example, two triangles, which will be congruent.

## Central symmetry in the Cartesian plane

The central symmetry, in the Cartesian plane, can be evidenced in the coordinates of the respective points. If the center of symmetry is then two points A (x1, y1) and B (x2, y2) are symmetric if:

x2 = -x1

y2 = -y2

That is, and (-4,3) are symmetric with respect to

However, the center of symmetry can be at any coordinate. Suppose we have two points A (x1, y1) and B (x2, y2). These are symmetric about point C (a, b) when we observe the following:

x2 = -x1 + 2a

y2 = -y1 + 2b

For example, (-4, -6) and are symmetric about the point.

## Central symmetry of polygons

As we described, the central symmetry can be fulfilled between two polygons. That is, when each point of one of them has a corresponding equidistant point in the other polygon, both being congruent (their sides and interior angles are of the same measure).

For example, we can see it in the following image:

Triangle ABC and triangle DEF are symmetric about the center of the Cartesian plane. And this can be evidenced by the coordinates of the vertices: A, B and C correspond to D (-4-2), E (-2, -6) and F (-10, -8), respectively.