Chapter 5 Properties of Triangles Perpendicular and Angle

BisectorsSec 5.1

Goal: To use properties of perpendicular bisectors and angle bisectors

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

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P

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

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

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P

AB CD $%%%%%%%%%%%%% %

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

Example

Does D lie on the perpendicular bisector of ? ?WhyAB

A

C

B

D

67

Example Is a perpendicular bisector of ? Why, or why not?CD

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A

C

B

D

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

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 %%%%%%%%%%%%%%%%%%%%%%%%%%%%

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

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

Examples Draw the segment that represents the distance indicated.

R M

T

P

L

A

R

R perpendicular to LM

T perpendicular to AP

Examples Name the segment whose length represents the distance between:

M to AT

T to HM

H to MT TH

AM