Download A median of a triangle is a segment whose endpoints are a vertex

Medians and Altitudes of Triangles
Objectives
• Use properties of medians of a
triangle
• Use properties of altitudes of a
triangle
Def: A median of a triangle is a segment whose endpoints are a
vertex and the midpoint of the side opposite that vertex.
A
B
D
BD is a median of  ABC
C
In your notebook, draw the three medians in the triangle.
P
Def: The centroid is the point of concurrency of the medians of a
triangle.
NOTE: The distance from the vertex to the centroid is twice the
distance from the centroid to the midpoint of the side opposite
that vertex.
y
2z
2x
z
x
The centroid is ALWAYS
inside the triangle.
2y
NOTE: The centroid is the center of balance (center of mass) or
balance point for the triangle.
Def: An altitude is a segment that represents the distance from a
vertex to the line containing the opposite side of the triangle.
A
Index Card
B
D
C
AD is one of the three altitudes of  ABC
Use your index card to draw the three altitudes of each triangle
on the sheet provided.
In each triangle do the altitudes intersect? The LINES containing
the altitudes of triangle do intersect at one point.
Def: The point of concurrency of the three altitudes (or the lines
containing the three altitudes) of a triangle is called the
orthocenter.
Where is the orthocenter with respect to each of the
following triangles?
Acute Triangle
Inside the triangle
Obtuse Triangle
Outside the triangle
Right Triangle
The vertex of the right
angle
The Nine Point Circle
© Copyright 2001, Jim Loy
In any triangle these nine points all lie on a circle (the green circle in the
diagram):
•The bases of the three altitudes.
•The midpoints of the three sides.
•The midpoints between the orthocenter (point where the three
altitudes meet) and each of the three vertices
Isosceles Triangles
and Special Segments
On the paper provided, draw the three medians, the three altitudes,
and the three perpendicular bisectors where indicated.
What do you notice as you compare the
three drawings????
NOTE: The median from the vertex angle of an isosceles triangle
is also an angle bisector, a perpendicular bisector, and an altitude
of the triangle.
The four points of concurrency all lie on this segment. (The
points of concurrency of an isosceles triangle are collinear.)
NOTE: The altitudes from the base angles are congruent.
The medians from the base angles are congruent.
What would be true about any median of an equilateral triangle???
Any median of an equilateral triangle is also an angle bisector, an
altitude, and a perpendicular bisector.
NOTE: The centroid, incenter, orthocenter, and circumcenter are
the same point in an equilateral triangle.