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perpendicular bisector
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(Definition)
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Let
 be a line segment in a plane (we are assuming the Euclidean plane). A bisector of
is any line that passes through the midpoint of
. A perpendicular bisector of
is a bisector that is perpendicular to
.
It is an easy exercise to show that  is a perpendicular bisector of
iff every point lying on is equidistant from  and  . 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
using the standard ruler and compass construction is as follows:
- use a compass to draw the circle
 centered at point with radius the length of
, by fixing one end of the compass at and the movable end at ,
- similarly, draw the circle
 centered at with the same radius as above, with the compass fixed at and movable at ,
- and intersect at two points, say
 (why?)
- with a ruler, draw the line
 ,
- then is the perpendicular bisector of
.
Figure: The construction of a perpendicular bisector
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(Note: we assume that there is always an ample supply of compasses and rulers of varying sizes, so that given any positive real number  , we can find a compass that opens wider than and a ruler that is longer than ).
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  in a Euclidean plane, let  be the unique circle determined by . Then the center of is located at the intersection of the perpendicular bisectors of
 and
 .
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"perpendicular bisector" is owned by CWoo. [ full author list (3) ]
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(view preamble)
Cross-references: collinear, center, opens, real number, positive, sizes, ample, ruler, intersect, fixed, length, radius, circle, compass, ruler and compass construction, point, iff, perpendicular, midpoint, passes through, line, Euclidean plane, plane, line segment
There are 28 references to this entry.
This is version 14 of perpendicular bisector, born on 2006-12-22, modified 2007-12-18.
Object id is 8652, canonical name is PerpendicularBisector.
Accessed 4364 times total.
Classification:
| AMS MSC: | 51N05 (Geometry :: Analytic and descriptive geometry :: Descriptive geometry) | | | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) | | | 51M15 (Geometry :: Real and complex geometry :: Geometric constructions) |
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Pending Errata and Addenda
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