Central Symmetry

Definition

Are two points M and O .
On says that is the symmetrical one for Me of M compared to O if, and only if, O is the medium of ME ''; i.e. MO = OM' and that M , O and is aligned to Me and distincts.
On says whereas O is the center of symmetry segment ME ''.

Constructions of symmetrical of a point compared to another point

With the rule and the compass

To plot the straight line ( MO ).
To trace the arc of circle of center O and MO .
the point of intersection of this arc of circle with the line ( MO ) is the point Me , symmetrical of the point M compared to O .

With the compass alone

To trace the arc of circle of center O and MO .
To trace the arc of circle of center M and ray 2x MO .
the point of intersection of these two arcs of circle is Me , the symmetrical one of M compared to O .

Properties

NB : Here, when we say " symétrique" , it is necessary to include/understand symmetrical compared to a point.

Property 1 : The symmetrical one of a line D is a line of which is parallel to D . That of a segment is a segment A' B' '' such as AB = A' B' .

Property 2 : The symmetrical one of a circle C of center O and ray R is a circle It of center O' , the symmetrical one of O , and of the same ray R .

Property 3 known as " of conservation" : Central symmetry preserves:

lengths;
angles (the symmetrical one of an angle is an of the same angle measures);
parallels (the symmetrical ones of two parallel straight lines are parallel);
surfaces (the symmetrical one of a figure is a of the same figure surface).

See too

axial Symmetry (elementary mathematics);
Symmetry (geometrical transformation).

Random links: