GEOMETRY: Chapter 5
5.3: Medians and Altitudes
A median of a triangle is a segment from
a vertex to the midpoint of the opposite
side. The three medians of a triangle are
concurrent. The point of concurrency,
called the centroid, is inside the triangle.
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 319.
Theorem 5.7: Concurrency of Medians of a Triangle
The medians of a triangle
intersect at a point that is
two thirds of the distance
from each vertex to the
midpoint of the opposite
side.
The medians of ABC meet at P and
2
2
2
AP  AE , BP  BF , and CP  CD.
3
3
3
Image taken from: Geometry. McDougal Littell: Boston, 2007. P. 319.
Ex. 1: In triangle HJK, P is the centroid and JP=12.
Find PT and JT.
Answer: 6,18
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 320.
Ex. 2: The vertices of triangle ABC are
A(1,5), B(5,7), and C(9,3).
Which ordered pair give the coordinates
of the centroid of triangle ABC?
A.(3,6)
B.(5,4)
C.(5,5)
D.(7,5)
Answer: C
Altitudes: An altitude of a triangle is the
perpendicular segment from a vertex to the
opposite side or to the line that contains the
opposite side.
Concurrency of Altitudes—the point at which
the lines containing the three altitudes of a
triangle intersect is called the orthocenter of
the triangle.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 320.
Theorem 5.8:
Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a triangle
are concurrent.
The lines containing AF , BE, and CD meet at G.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 320.
Ex. 3: Show that the orthocenter can be inside,
on, or outside the triangle.
Ex. 3: Show that the orthocenter can be inside,
on, or outside the triangle.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 320.
Ex. 4: Prove that if an angle bisector of a
triangle is also an altitude, then the triangle
is isosceles.
Given: ABC , with BD as an angle bisector
and altitude to AC.
Prove: ABC is isosceles.
Images taken from: Geometry. McDougal Littell: Boston, 2007. P. 321.
Write proof for Ex. 4 here:
5.3, p. 282, #3-10 all, 17-20 all.