Perpendicular & AnglePerpendicular & AngleBisectorsBisectors

Objectives

Identify and use ┴ bisectors and bisectors in ∆s

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

Example

What is the length of AB?

BCBA

1064 xx

5x

)5(4BA

20BA

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 ∆.

Example

What is the length of RM?

RPRM 527 xx

5x

)5(7RM

35RM

Your Turn

What is the length of FD?

FDBF

9436 xx3x

9)3(4 FD

21FD