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Medians, Altitudes and Angle Bisectors

- Every triangle has
- 1. 3 medians,
- 2. 3 angle bisectors and
- 3. 3 altitudes.

B

A

C

- Given ?ABC, identify the opposite side
- of A.
- of B.
- of C.

Any triangle has three medians.

B

L

M

A

N

C

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.

Any triangle has three angle bisectors.

B

E

F

A

C

D

M

Note An angle bisector and a median of a

triangle are sometimes different.

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 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

Can a side of a triangle be its altitude?

YES!

A

G

C

B

RIGHT

If ?ABC is a right triangle, identify its

altitudes.

D

B

C

If BD DC, then we say that

D is equidistant from B and C.

- Definition of an Equidistant Point
- A point D is equidistant from B and C if and

only if BD DC.

T

V

M

R

S

U

RT TS RV VS RU US

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

- Theorem If a point lies on the perpendicular

bisector of a segment, then the point is

equidistant from the endpoints of the segment.

- Theorem If a point lies on the perpendicular

bisector of a segment, then the point is

equidistant from the endpoints of the segment.

The converse of this theorem is also true

Theorem If a point is equidistant from the

endpoints of a segment, then the point lies on

the perpendicular bisector of the segment.

H

F

G

Given HF HG Conclusion H lies on the

perpendicular bisector of FG.

T

V

R

S

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.

- Definition of a Distance Between a Line and a

Point not on the Line - The distance between a line and a point not on

the line is the length of the perpendicular

segment from the point to the line.

B

Let AD be a bisector of ?BAC, P lie on AD, PM ?

AB at M, NP ? AC at N.

M

P

A

N

C

- Theorem If a point lies on the bisector of an

angle, then the point is equidistant from the

sides of the angle.

Theorem If a point lies on the bisector of an

angle, then the point is equidistant from the

sides of the angle.

The converse of this theorem is not always true.

Theorem If a point is in the interior of an

angle and is equidistant from the sides of the

angle, then the point lies on the bisector of the

angle.

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