MEDIANS AND ALTITUDES OF TRIANGLES

(SPECIAL SEGMENTS)

Definitions and Theorems…..•Median-- A segment whose

endpoints are a vertex of the triangle and the midpoint of the opposite side.

• Centroid-- The point where the three medians of a triangle intersect.

Definitions and Theorems…..

•Centroid Theorem-- 3

2The centroid is located the distance from the vertex to the midpoint

on the opposite side.

A B

C

D

X

Y

Z 3

3AZ = of AZ ZD = of AZ3

2

AD = of AZ3

1

Centroid

Centroid ExampleUsing Triangle ABC, find the segment lengths of AG and CE.

A

B

CD

EFG

AF = 9AG = GF =

EG = 2.4 GC = EC =

BG = 8 GD = BD =

Construct three Medians

Q (0,8)

R (6,4)

P (2,0)

Centriod (2½, 4)

Definitions and Theorems…..

•Altitude--

• Orthocenter-- The point where the three altitudes of a triangle intersect.

Is a perpendicular segment from the vertex to the opposite side of the triangle.

Altitudes---height•Altitudes of a Triangle-

A B

C

D

X

Y

Z

• Orthocenter

• The orthocenter can be located inside, outside or on the given triangle

Construct three altitudes

m =

R (6,4)

P (2,0)

Q (0,8)

m =

m =

m =

m = m =

3

2

1

1

1

42

3

1

14

1

Orthocenter (4.5, 3.5)

Steps for Constructing Special Segments:

•1. Slide 1 --- Steps for Constructing Perpendicular Bisectors

•2. Slide 2 --- Steps for Constructing Angle Bisectors

•3. Slide 3 --- Steps for Constructing medians and altitudes

Perpendicular Bisectors

1.Graph the points2.Find the midpoint of each side3.Plot the midpoints4.Find the slope of each side5.Find the perpendicular slope of each side6.From the midpoint, count using the perpendicular slope and plot another point7.Draw a line segment connecting the midpoint and the point8.The point where all 3 perpendicular bisectors cross is called the circumcenter

You will need a straight edge for this construction

Angle Bisectors

1.Graph the points and draw a triangle2.Using a compass, draw an arc using one vertex as the center. This arc must pass through both sides of the angle3.Plot points where the arc crosses the sides of the angle4.From the points on the sides, create two more arcs (with the same radius) that cross. Plot a point where the two arcs cross5.Draw a line segment from the vertex to the point where the small arcs cross.6.When bisecting angles on a triangle, the point where all three angle bisectors cross is called the incenter

You will need a compass and a straight edge for this construction

Medians

1.Graph the points for the triangle2.Find the midpoint of each side3.Draw a line segment connecting the midpoint to the opposite vertex4.The point where the 3 medians cross is called the centroid

Altitudes

1.Graph the points for the triangle2.Find the slope of each side3.Find the perpendicular slope of each side4.From the opposite vertex, count using the perpendicular slope. Plot another point and draw a line segment to connect the vertex to this point5.The point where all 3 altitudes cross is called the orthocenter

You will need a straight edge for both of these constructions