Name: ___________________
Ms. Anderle
Date: ______ Period:_____
Review Sheet: Quest
Review Sheet:
Points of Concurrency
Points of Concurrency Constructions:
A point of concurrency is the point where three lines intersect. In a triangle,
there are four points of concurrency. Each one of these points can be found
through a different construction.
1) Centroid: the point of concurrency of the medians of a triangle. It is the
center of gravity of a triangle. In order to construct the centriod,
construct the medians of each side of the triangle.
2) Orthocenter: the point of concurrency of the altitudes of a triangle. In
order to construct the orthocenter, construct a perpendicular line from a
point off the line on each side of the triangle. The point off the line will be
a vertex of triangle.
a. Altitude: in a triangle, it is a line drawn from a vertex of a triangle
and forms a 90° angle with the opposite side.
3) Incenter: the point of concurrency of the angle bisectors of a triangle. The
incenter is equidistant from each side of the triangle. In order to
construct the incenter, construct an angle bisector of each side of the
triangle. Draw the angle bisector from the angle, to the opposite side of
the triangle.
4) Circumcenter: the point of concurrency of the perpendicular bisectors of
the sides of the triangle. The circumcenter is equidistant from the
vertices of a triangle. In order to construct the circumcenter, construct
the perpendicular bisector of each side of the triangle.
Points of Concurrency Word Problems:
Other types of problems that we see when dealing with points of concurrency in
triangles are word problems. Two that we have learned involve the centroid and
the circumcenter.
Circumcenter Word Problems:
Note: The circumcenter is equidistant (the same distance) from each vertex of
the triangle. This means that all the segments drawn from the vertex of the
triangle to the circumcenter are the same exact distance (have the same value).
Example: The perpendicular bisectors of ▲ABC intersect at P. If AP = 5 + x, BP
= 10 and CP = 2y, find x and y.
In order to solve this problem, we need to realize that
AP = BP = CP. Therefore, we can set each one of them equal to each
other. In this problem it makes the most sense to set AP = BP and CP =
BP.
5 + x = 10
2y = 10
-5
-5
2
2
x=5
y=5
***To find other similar problems, review the worksheets that I handed out and
the problems that were discussed in class.
Centroid Word Problems:
If drawn correctly, the medians will intersect at the same point, called the centroid.
B
F
A

P
D
E
C
The medians of the triangle are concurrent at point P
P is called the centroid
The three medians divide the triangle into six regions of equal area
Also the centroid divides the medians into two segments in the ratio
AP 2
CP 2
BP 2



and
and
PD 1
PF 1
PE 1
The centroid is exactly two-thirds the way along each median
AP BP CP 2



AD BE CF 3
2
, such that
1
2
(AD) = AP
3
Notice that AP, BP, and CP are the parts
2
(BE) = BP
3
of the medians drawn from the vertex
to the centroid
2
(CF) = CP
3
The centroid is located one-third of the perpendicular distance between each side and the
opposing point
PD PE PF 1



AD BE CF 3
In order to solve certain centriod word problems, we use these ratio
relationships.
Example: In QRS with medians QA and SC are concurrent at point P. If QA =
15, find QP.
R
2/3 = QP/QA
2/3 = x/15
C
P
A
30 = 3x
x = 10
Q
S
Another type of centroid word problem is one that deals with finding the
coordinates of the centroid.
Given the three vertices of a triangle, (x1,y1), (x2,y2), and (x3,y3),
x1  x2  x3 y1  y2  y3
,
the coordinates of the centroid =
3
3
F
G
H
IJ
K
Example: Given  ABC with coordinates A(0,0), B(4,0), and C(2,6), find the coordinates of
the centroid.
centroid =
x x x y y y I
F
G
H 3 , 3 J
K
1
2
3
1
2
3
centroid = (2, 2)
***To find other similar problems, please refer to the problems that were given
in class and handouts.
Good Luck on the Quest!!!!