Challenge Problems for 3.1-3.3

Challenge Problems for 3.1-3.3
Remember points of concurrency from geometry? If you haven’t completely blocked that part of your mathematical education
from your memory, you may recall that three or more lines are concurrent if they all intersect at the same point. The point that
they all have in common is called the point of concurrency. What we discovered in geometry is that there are a number of
points of concurrency associated with triangles: Centroid (where the medians intersect), Circumcenter (where the
perpendicular bisectors intersect), Incenter (where the angle bisectors intersect), and Orthocenter (where the altitudes
intersect).
Points of Concurrency for Triangles
Circumcenter
Incenter
Centroid



Intersection of the medians
Median: segment that
connects a vertex to the
midpoint of the opposite side

Point C

Intersection of the
perpendicular bisectors
Perp. Bisector: line that
intersects a side at its midpoint
and is perpendicular to it

Point C2


Intersection of the angle
bisectors
Angle Bisector: segment that
bisects an angle

Point I
Orthocenter



Intersection of the altitudes
Altitude: a segment that starts
at a vertex and is
perpendicular to a line
containing the other side of
the triangle
Point O
I
C2
O
C
Find the indicated point of concurrency.
1.
Find the equations of the three medians for the triangle below. Then find the coordinates of the Centroid.
6
4
2
5
5
2
2.
Find the equations of the three perpendicular bisectors of the sides of the triangle below. Then find the coordinates for the
Circumcenter.
6
4
2
5
2
4
6
10
3.
Find the equations of the three altitudes for the triangle below. Then find the coordinates for the Orthocenter.
10
8
6
4
2
5
5
2