chapter 5 properties of triangles perpendicular and angle bisectors sec 5.1 goal: to use properties...
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Chapter 5 Properties of Triangles Perpendicular and Angle
BisectorsSec 5.1
Goal: To use properties of perpendicular bisectors and angle bisectors
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Perpendicular BisectorPerpendicular Bisector – a segment, ray, line, or plane that is perpendicular to a segment at its midpoint.
A
C
B
M
If is a perpendicular bisector of AB, then AP PB and AB
CPCP
$%%%%%%%%%%%%% %
$%%%%%%%%%%%%% %
P
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EquidistantEquidistant from two points means that the distance from each point is the same.
A
C
B
C is equidistant from A and B;
therefore, CA = CB
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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.
A
C
B
DIf is the perpendicular bisector of AB,
then CA CB. Likewise, DA DB.
CD
$%%%%%%%%%%%%% %
P
AB CD $%%%%%%%%%%%%% %
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Converse of the Perpendicular Bisector
TheoremConverse 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.
A
C
B
D
If DA DB, then D lies on the perpendicular bisector of AB.
P
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Example
Does D lie on the perpendicular bisector of ? ?WhyAB
A
C
B
D
67
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Example Is a perpendicular bisector of ? Why, or why not?CD
$%%%%%%%%%%%%% %AB
A
C
B
D
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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
P
m
The distance from point P to line m is the length of QP
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Angle Bisector TheoremAngle Bisector Theorem – If a point (D) is on the bisector of an angle, then it is equidistant from the two sides of the angle.
A
C
B
DIf , is the angle bisector of BAC,
then BD CD
AD
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BD AB and DC AC %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Converse of the Angle Bisector Theorem
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.
A
C
B
D
If DB DC,
then AD lies on the bisector of BAC
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Examples
Does the information given in the diagram allow you to conclude that C is on the perpendicular bisector of AB?
A
A
B
BC
C
A
B
CP
P
D
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ExamplesDoes the information given in the diagram allow you to conclude that P is on the angle bisector of angle A?
P
6
P
P6
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Examples Draw the segment that represents the distance indicated.
R M
T
P
L
A
R
R perpendicular to LM
T perpendicular to AP
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Examples Name the segment whose length represents the distance between:
M to AT
T to HM
H to MT TH
AM