1. Medians and Altitudes of Triangles (Special Segments)

2. 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.

3. Definitions and Theorems…..
The centroid is located
the distance from the
vertex to the midpoint on the opposite side.
Centroid Theorem-- Y B
Centroid A D
AZ = of AZ
ZD = of AZ Z X C
AD = of AZ

4. Centroid Example BG = 8 EG = 2.4 AF = 9 GD = GC = AG = BD = EC = GF =
Using Triangle ABC, find the segment lengths of AG and CE.
BG = 8
GD =
BD =
EG = 2.4
GC =
EC =
AF = 9
AG =
GF = B E F G A D C

5. Construct three Medians
Q (0,8)
Centriod (2½, 4)
R (6,4)
P (2,0)

6. Definitions and Theorems…..
Is a perpendicular segment from the vertex to the opposite side of the triangle.
Altitude--
Orthocenter--
The point where the three altitudes of a triangle intersect.

7. Altitudes---height Altitudes of a Triangle- Orthocenter
The orthocenter can be located inside, outside or on the given triangle Y B A D Z X C

8. Construct three altitudes
Orthocenter (4.5, 3.5)
Q (0,8)
m =
m =
m =
R (6,4)
m =
m =
m =
P (2,0)

9. Steps for Constructing Special Segments:
1. Slide Steps for Constructing Perpendicular Bisectors
2. Slide Steps for Constructing Angle Bisectors
3. Slide Steps for Constructing medians and altitudes

10. Perpendicular Bisectors Graph the points
Find the midpoint of each side
Plot the midpoints
Find the slope of each side
Find the perpendicular slope of each side
From the midpoint, count using the perpendicular slope and plot another point
Draw a line segment connecting the midpoint and the point
The point where all 3 perpendicular bisectors cross is called the circumcenter
You will need a straight edge for this construction

11. Graph the points and draw a triangle
Angle Bisectors
Graph the points and draw a triangle
Using a compass, draw an arc using one vertex as the center. This arc must pass through both sides of the angle
Plot points where the arc crosses the sides of the angle
From the points on the sides, create two more arcs (with the same radius) that cross. Plot a point where the two arcs cross
Draw a line segment from the vertex to the point where the small arcs cross.
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

12. Graph the points for the triangle Find the midpoint of each side
Medians
Graph the points for the triangle
Find the midpoint of each side
Draw a line segment connecting the midpoint to the opposite vertex
The point where the 3 medians cross is called the centroid
Altitudes
Graph the points for the triangle
Find the slope of each side
Find the perpendicular slope of each side
From the opposite vertex, count using the perpendicular slope. Plot another point and draw a line segment to connect the vertex to this point
The point where all 3 altitudes cross is called the orthocenter
You will need a straight edge for both of these constructions