Medians, Medians, Altitudes and Altitudes and

Angle BisectorsAngle Bisectors

Every triangle has Every triangle has

1. 1. 3 medians,3 medians,

2. 2. 3 angle bisectors and3 angle bisectors and

3. 3. 3 altitudes.3 altitudes.

A

B

C

Given ABC, identify the opposite side

1. of A.

2. of B.

3. of C.

BC

AC

AB

Any triangle has three medians.

A

B

C

L

M

N

Let L, M and N be the midpoints of AB, BC and AC respectively.Hence, CL, AM and NB are medians of ABC.

Definition of a Median of a Triangle

A median of a triangle is a segment whose endpoints are a vertex of a triangle and a midpoint of the side opposite that vertex.

A

B

CD

E F

In the figure, AF, DB and EC are angle bisectors of ABC.Definition of an Angle Bisector of a Triangle

A segment is an angle bisector of a triangle if and only if

a) it lies in the ray which bisects an angle of the triangle and

b) its endpoints are the vertex of this angle and a point on the opposite side of that vertex.

Any triangle has three angle bisectors.

Note: An angle bisector and a median of a triangle are sometimes different.

BM is a median and BD is an angle bisector of ABC.

M

Let M be the midpoint of AC.

Any triangle has three altitudes.

Definition of an Altitude of a Triangle

A segment is an altitude of a triangle if and only if it has one endpoint at a vertex of a triangle and the other on the line that contains the side opposite that vertex so that the segment is perpendicular to this line.

ACUTE OBTUSE

B

A

C

RIGHT

A

B C

If ABC is a right triangle, identify its altitudes.

BG, AB and BC are its altitudes.

G

Can a side of a triangle be its altitude?YES!

Definition of an Equidistant PointDefinition of an Equidistant Point

A point D is A point D is equidistantequidistant from B and C from B and C if and only if BD = DC.if and only if BD = DC.

D

BC

If BD = DC, then we say that

D is equidistant from B and C.

Theorem:Theorem: If a point lies on the If a point lies on the perpendicular perpendicular bisector of a segment, bisector of a segment, then the point is then the point is equidistant from the equidistant from the endpoints of the endpoints of the segment.segment.

R S

T

V

U

Let TU be a perpendicular bisector of RS.

Then, what can you say about T, V and U?

RT = TS

RV = VS

RU = US

M

Theorem:Theorem: If a point lies on the If a point lies on the perpendicular perpendicular bisector of a segment, bisector of a segment, then the point is then the point is equidistant from the equidistant from the endpoints of the endpoints of the segment.segment.

The converse of this theorem is also true:

Theorem:Theorem: If a point is equidistant from If a point is equidistant from the endpoints of a segment, then the the endpoints of a segment, then the point lies on the perpendicular bisector point lies on the perpendicular bisector of the segment.of the segment.

F G

H

Given: HF = HG

Conclusion: H lies on the perpendicular bisector of FG.

Theorem: If two points and a segment lie on the same plane and each of the two points are equidistant from the endpoints of the segment, then the line joining the points is

the perpendicular bisector of the segment.

R S

T

V

If T is equidistant from R and S and similarly, V is equidistant from R and S, then what can we say about TV?

TV is the perpendicular bisector of RS.

Definition of a Distance Between a Line Definition of a Distance Between a Line and and a Point not on the Linea Point not on the Line

The distance between a line and a The distance between a line and a point not on the linepoint not on the line is the length of is the length of the perpendicular segment from the the perpendicular segment from the point to the line.point to the line.

D

Theorem:Theorem: If a point lies on the bisector If a point lies on the bisector of an angle, then the point is of an angle, then the point is equidistant from the sides of the equidistant from the sides of the angle.angle.

Let AD be a bisector of BAC,

P lie on AD,

PM AB at M,

NP AC at N.

A

B

C

M

N

P

Then P is equidistant from AB and AC.

Theorem:Theorem: If a point lies on the bisector If a point lies on the bisector of an angle, then the point is of an angle, then the point is equidistant from the sides of the equidistant from the sides of the angle.angle.

The converse of this theorem is not always true.

Theorem:Theorem: If a point is in the interior of If a point is in the interior of an anglean angle and is equidistant from the and is equidistant from the sides of the angle, then the point lies sides of the angle, then the point lies on the bisector of the angle.on the bisector of the angle.