Post on 24-Dec-2015
Perpendicular Bisector
A ┴ bisector of a ∆ is
a line, segment, or
ray that passes
through the midpoint
of one of the sides of
the ∆ at a 90° .
Side AB
perpendicular bisector
PA B
C
┴ Bisector Theorems
Theorem 5.1 – Any point on the ┴ bisector of a segment is equidistant from the endpoints of the segment.
Theorem 5.2 – Any point equidistant from the endpoints of a segment lies on the ┴ bisector of the segment.
┴ Bisector Theorems (continued)
Basically, if CP is the perpendicular bisector of AB, then PA ≅ PB.
Side AB
perpendicular bisector
PA B
C
┴ Bisector Theorems (continued)
Since there are three sides in a ∆, then there are three ┴ Bisectors in a ∆.
These three ┴ bisectors in a ∆ intersect at a common point called the circumcenter.
┴ Bisector Theorems (continued)
Theorem 5.3 (Circumcenter Theorem) The circumcenter of a ∆ is equidistant from the vertices of the ∆.
Notice, a circumcenter of a ∆ is the center of the circle we would draw if we connected all of the vertices with a circle on the outside (circumscribe the ∆).
circumcenter
Angle Bisectors of ∆s Another special bisector which we have already
studied is an bisector. As we have learned, an bisector divides an into two parts. In a ≅ ∆, an bisector divides one of the ∆s s into two ≅ s.(i.e. if AD is an bisector then BAD ≅ CAD)
B D C
Angle Bisectors of ∆s (continued) Theorem 5.4 (Angle Bisector Theorem) – Any
point on an bisector is equidistant from the sides of the .
Theorem 5.5 (Converse of the Angle Bisector Theorem) – Any point equidistant from the sides of an lies on the bisector.
Angle Bisectors of ∆s (continued)
As with ┴ bisectors, there are three bisectors in any ∆. These three bisectors intersect at a common point we call the incenter.
incenter
Angle Bisectors of ∆s (continued)
Theorem 5.6 (Incenter Theorem) The incenter of a ∆ is equidistant from each side of the ∆.