perpendicular bisector

Let AB¯ be a line segmentMathworldPlanetmath in a plane (we are assuming the Euclidean planeMathworldPlanetmath). A bisectorMathworldPlanetmath of AB¯ is any line that passes through the midpointMathworldPlanetmathPlanetmathPlanetmath of AB¯. A perpendicular bisector of AB¯ is a bisector that is perpendicularMathworldPlanetmathPlanetmathPlanetmath to AB¯.

It is an easy exercise to show that a line is a perpendicular bisector of AB¯ iff every point lying on is equidistant from A and B. From this, one concludes that the perpendicular bisector of a line segment is always unique.

A basic way to construct the perpendicular bisector given a line segment AB¯ using the standard ruler and compass construction is as follows:

  1. 1.

    use a compass to draw the circle C1 centered at point A with radius the length of AB¯, by fixing one end of the compass at A and the movable end at B,

  2. 2.

    similarly, draw the circle C2 centered at B with the same radius as above, with the compass fixed at B and movable at A,

  3. 3.

    C1 and C2 intersect at two points, say P,Q (why?)

  4. 4.

    with a ruler, draw the line PQ=,

  5. 5.

    then is the perpendicular bisector of AB¯.

Figure 1: The construction of a perpendicular bisector

(Note: we assume that there is always an ample supply of compasses and rulers of varying sizes, so that given any positive real number r, we can find a compass that opens wider than r and a ruler that is longer than r).

One of the most common use of perpendicular bisectors is to find the center of a circle constructed from three points in a Euclidean plane:

Given three non collinear points X,Y,Z in a Euclidean plane, let C be the unique circle determined by X,Y,Z. Then the center of C is located at the intersection of the perpendicular bisectors of XY¯ and YZ¯.

Title perpendicular bisector
Canonical name PerpendicularBisector
Date of creation 2013-03-22 16:29:03
Last modified on 2013-03-22 16:29:03
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 18
Author CWoo (3771)
Entry type Definition
Classification msc 51M15
Classification msc 51N20
Classification msc 51N05
Synonym center normal
Related topic CircumcircleMathworldPlanetmath
Defines bisector