Perpendicular bisector

The mean perpendicular or bisector or (Austrian) bisector is a special Straight which in plane geometry is studied. A generalization to three dimensions is the perpendicular plane .

definition

The perpendicular is the amount of all points , the two given points each the same distance have:

{\ displaystyle s = \ left \ {X \ mid {\ overline {XA}} = {\ overline {XB}} \ right \}}

Another possible definition is: The mid-perpendicular is the set of centers of all circles that go through two given points.

The mid-perpendicular is thus a straight line that is orthogonal (that is, perpendicular) to the line connecting the two points and goes through their center.

construction

Construction of a line symmetrical

You construct a perpendicular line between two given points and by drawing arcs around these two points with a pair of compasses with the same radius , which must be greater than half the distance between the two points. The two points of intersection of these two circular lines define a straight line. This straight line is the center perpendicular of the route . ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle AB}$

Calculation in the coordinate system

Are in a two-dimensional Cartesian coordinate system , two points and having given this is the linear equation of the perpendicular bisector: ${\ displaystyle A (x_ {A} | y_ {A})}$ ${\ displaystyle B (x_ {B} | y_ {B})}$ ${\ displaystyle y_ {A} \ neq y_ {B}}$

{\ displaystyle y = - {\ frac {x_ {A} -x_ {B}} {y_ {A} -y_ {B}}} x + {\ frac {x_ {A} ^ {2} -x_ {B} ^ {2} + y_ {A} ^ {2} -y_ {B} ^ {2}} {2 \ cdot (y_ {A} -y_ {B})}}}

If , then the (non-functional) equation reads: ${\ displaystyle y_ {A} = y_ {B}}$ ${\ displaystyle x = {\ frac {1} {2}} (x_ {A} + x_ {B})}$

Center perpendiculars in the triangle

The perpendiculars of a triangle intersect at a point, namely in the circumference of the center of the triangle. This perimeter goes through all corners of the triangle (see also: Distinguished points in the triangle ).

Vertical lines in an isosceles triangle, as it were, as a bisector ; the vertex is outside the plane of the drawing.

In the isosceles triangle , the vertical line for the angle at the apex of the two same legs can also fulfill the function of bisecting the angle . This is particularly advantageous in the case of isosceles triangles in which the vertex does not lie within the plane of the drawing.

Center plumb plane

The median perpendicular to two points and in space is the plane that is perpendicular to the connecting line and goes through the center of this line, i.e. the plane of symmetry of the points and . ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle [AB]}$ ${\ displaystyle M}$ ${\ displaystyle A}$ ${\ displaystyle B}$

In analytical geometry , an equation of the perpendicular plane in normal form is obtained by using the vector as the normal vector and the point (with the position vector ) as the starting point: ${\ displaystyle {\ vec {n}} = {\ vec {AB}}}$ ${\ displaystyle M}$ ${\ displaystyle {\ vec {m}}}$

{\ displaystyle {\ vec {n}} \ cdot \ left ({\ vec {x}} - {\ vec {m}} \ right) = 0}

literature

Rolf Baumann: Geometry . with exercises and solutions. Mentor, Munich 2002, chapter 3.1.
Cornelia Niederdrenk-Felgner: Lambacher-Schweizer . Mathematics textbook for 7th grade (G9) at high schools (Baden Württemberg). Klett, Stuttgart 1994, ISBN 3-12-731370-5 .

Web links

Wiktionary: mid-perpendicular - explanations of meanings, word origins, synonyms, translations