Medians, Altitudes and Concurrent Lines Section 5-3.

Medians, Medians, Altitudes and Altitudes and

Concurrent LinesConcurrent Lines Section 5-3Section 5-3

A

B

C

Given ABC, identify the opposite side

of A

of B

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.

Properties of Medians

The median starts at a vertex and ends

at the midpoint of the opposite side.

Centroid

Properties of Medians

Properties of Medians

Centroid of a Triangle: The point of concurrency of the medians of a triangle.

The medians of a triangle are concurrent at a The medians of a triangle are concurrent at a point that is point that is two thirdstwo thirds the distance from the distance from each vertex to the midpoint of the opposite each vertex to the midpoint of the opposite side.side.

This point of intersection is called a This point of intersection is called a centroidcentroid..

D

G

F

C

J

H

EDC = 2/3(DJ)EC = 2/3(EG)FC = 2/3(FH)

Theorem about Medians

The The centroidcentroid is 2/3’s of the distance is 2/3’s of the distance from the vertex to the side.from the vertex to the side.

2x2x

xx

1010

55

3232

XX1616

Properties of Medians

Properties of Medians

In the figure below, In the figure below, DEDE = 6 = 6 and and ADAD = 16 = 16. . Find Find DBDB and and AFAF..

F

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 Any triangle has three angle bisectors.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.

Properties of Angle Bisectors

Angle bisectors start at a vertexAngle bisectors start at a vertexand and bisect the angle.bisect the angle.

IncenterIncenter

Properties of Angle Bisectors

Any point Any point on an angle on an angle bisector is bisector is equidistancequidistance from the e from the sides of the sides of the

angle angle

Properties of Angle Bisectors

This makes This makes the the IncenterIncenter an anequidistancequidistance from all 3 e from all 3 sidessides

Properties of Angle Bisectors

Any triangle has Any triangle has three (3)three (3) altitudes.altitudes.Definition of an Altitude of a TriangleDefinition of an Altitude of a Triangle

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

ACUTE OBTUSE

B

A

C

Properties of Altitudes

Start at a vertex and form a Start at a vertex and form a 9090° ° angleangle

with the line containing the with the line containing the opposite side.opposite side.OrthocenterOrthocenter

Properties of Altitudes

The The orthocenterorthocenter can be located can be locatedin the triangle, on the triangle orin the triangle, on the triangle oroutside the triangle.outside the triangle.

RightRight

Legs are altitudesLegs are altitudes

ObtuseObtuse

Properties of Altitudes

RIGHT

A

B C

If If ABC is a right triangle, identify its altitudes.ABC is a right triangle, identify its altitudes.

BG, AB and BC are its altitudes.BG, AB and BC are its altitudes.

G

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

Properties of Altitudes

MMedian goes from vertex to edian goes from vertex to mmidpointidpoint of segment of segment

opposite.opposite.

Altitude is Altitude is a perpendicular a perpendicular segment segment from vertex to from vertex to

segment opposite.segment opposite.

Compare Medians & Altitudes

Altitude ..Altitude .. Vertex .. 90Vertex .. 90°° .. .. OrthocenterOrthocenter

Angle BisectorAngle Bisector.... Angle into 2 equal angles .. Angle into 2 equal angles ..

IncenterIncenterPerpendicular BisectorPerpendicular Bisector……

90° .. bisects side .. 90° .. bisects side .. CircumcenterCircumcenterMedianMedian .. ..

Vertex .. Midpoint of side ..Vertex .. Midpoint of side ..CentroidCentroid

Give the best name for ABGive the best name for ABAA

BB

AA

BB

AA

BB

AA

BB

AA

BB||||

|| ||||||

MedianMedian Altitude Altitude None None Angle Angle PerpPerp Bisector Bisector BisectorBisector

Concurrency

Concurrent Lines: Three or more lines that meet at one point.

Point of Concurrency: The point at which concurrent lines meet.

l

m

n

P

k

Properties of Bisectors

Theorem 5-6: The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

Circumcenter of the Triangle: The point of concurrency of the perpendicular bisectors of a triangle.

Properties of Bisectors

Theorem 5-7: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

Incenter of the Triangle: The point of concurrency of the angle bisectors of a triangle.

Sum It Up

Figure

concurrent at.. which is…

bisector circumcenter

incenter

centroid

orthocenter

median

bisector

altitude

equidistant from vertices

equidistant from sides

2/3 distance from vertices to midpoint

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