Axial Symmetry

preliminary Note : one also speaks about orthogonal symmetry .

Definition

That is to say a point M and a line D .
On says that the point is the symmetrical one for Me of M compared to D if, and only if, D is the Médiatrice of ME ''.
It is said whereas D is the axis of symmetry segment ME ''.

Constructions of symmetrical of a point compared to a line

With the scale and the square

To plot the straight line passing by M and perpendicular to D .
Is O the point of intersection of this line and D .
Placer on ( MO ) the point Me such as MO = OM' .

Then, is the symmetrical one for Me of M compared to D .

With the compass alone

To trace an arc of circle of center M and unspecified, but sufficiently large ray to cut the line D in two points has and B .
Tracer the arc of circle of center has and of.
To trace the arc of circle of center B and.
These two arcs of circle cut in a point Me , which is the symmetrical one of M compared to D .

Properties

What follows is false for axial symmetry… to correct it!!

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

Property 1 : It is said that two figures are symmetrical when they are superimposed after folding along the line ( D ).

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" : Axial 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)
perimeters (the symmetrical one of a figure is a of the same figure perimeter)

See too

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

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