Chapter 5 Properties of Triangles
Perpendicular and Angle Bisectors
Sec 5.1
Goal:
To use properties of perpendicular
bisectors and angle bisectors
Perpendicular Bisector
Perpendicular Bisector – a segment, ray, line, or plane that is
perpendicular to a segment at its midpoint.
C

A

M
P
If CP is a perpendicular bisector of AB,
then AP  PB and CP  AB

B
Equidistant
Equidistant from two points means that the distance from each
point is the same.
C

A
C is equidistant from A and B;
therefore, CA = CB

B
Perpendicular Bisector Theorem
Perpendicular Bisector Theorem – If a point is on the perpendicular
bisector of a segment, then it is equidistant from the endpoints of
the segment.
C

A
If CD is the perpendicular bisector of AB,
then CA  CB. Likewise, DA  DB.
CD  AB


P
D

B
Converse of the Perpendicular
Bisector Theorem
Converse of the Perpendicular Bisector Theorem – If a point is
equidistant from the endpoints of a segment, then it is on the
perpendicular bisector of a segment.
C

A

P

B
 D
If DA  DB, then D lies on the perpendicular bisector of AB.
Example
Does D lie on the perpendicular bisector of AB ? Why ?
C


A
B
7

6
D
Example
Is CD a perpendicular bisector of AB ? Why, or why not?
C

A

D

B
Distance from a point to a line
The distance from a point to a line is defined to be the
shortest distance from the point to the line. This
distance is the length of the perpendicular segment.
 Q

m
P
The distance from point P to line m is the length of QP
Angle Bisector Theorem
Angle Bisector Theorem – If a point (D) is on the bisector of an
angle, then it is equidistant from the two sides of the angle.
B
If AD, is the angle bisector of BAC,
then BD  CD
A


C
BD  AB and DC  AC
D
Converse of the Angle Bisector Theorem
If DB  DC,
then AD lies on the bisector of BAC
B
A

D

C
Converse of the Angle Bisector Theorem – If a point is on the
interior of an angle, and is equidistant from the sides of the angle,
then it lies on the bisector of the angle.
Examples
Does the information given in the diagram allow you to
conclude that C is on the perpendicular bisector of AB?


A

B
A
D
C
P
B

A
C
C
P

B
Examples
Does the information given in the diagram allow you to
conclude that P is on the angle bisector of angle A?
 P
6
P
6
P
Examples
Draw the segment that represents the distance indicated.
L
R
T
M
R perpendicular to LM
R
P
A
T perpendicular to AP
Examples
Name the segment whose length represents the distance
between:
M to AT
M
A
T to HM
H to MT
H
T