Name ____________________________
Period ___________
Date _______
Geometry – Points of Concurrency of a Triangle – 5.4 Centroid & Orthocenter
1. You are using the same coordinate points as you used for your triangle when you found the
circumcenter!
2. Materials needed
 Graph paper (you can print additional graph paper from my website)
 A straightedge to draw ALL LINES
 A protractor may be helpful, but is not necessary
 A calculator
3. The Centroid – You will find the coordinates of the centroid of your triangle both by drawing a
graph and estimating and by using algebra to find the exact coordinates.
a. On a coordinate plane, plot your assigned coordinate points and draw your triangle. Make sure
to label your vertices with A, B, and C and include their coordinates on the graph.
b. Carefully draw the three medians of this triangle. Note: Since we are drawing, not constructing,
you must calculate the coordinates of the midpoints of each side using the midpoint formula (do
not measure!). Label the midpoints as follows: the midpoint of AB is D, the midpoint of BC is E,
and the midpoint of AC is F.
c. Label the centroid J and estimate its coordinates from your graph.
d. Calculate the coordinates of the centroid algebraically. In order to get full credit, you must show
all work in an organized way on a separate sheet of paper.

Find the equations of two of the medians. Please write the final equation of each line on the
correct median on the graph.

Calculate the point of intersection of the two lines. Your solution to this system of linear
equations will be the coordinates of the centroid. Compare your calculated centroid with
your estimated centroid. They should be reasonably close. Otherwise, you will need to
double check your work and try again.

Your calculations may involve fractions/decimals. . Use the exact coordinates, if possible.
If you must round, round your final coordinates to the nearest hundredth.
4. The Orthocenter – You will find the coordinates of the orthocenter of your triangle both by drawing
a graph and estimating and by using algebra to find the exact coordinates.
a. On a coordinate plane, plot your assigned coordinate points and draw your triangle. Make sure
to label your vertices with A, B and C and include their coordinates on the graph.
b. Carefully draw the three altitudes of the triangle. To make sure that the altitude is perpendicular
to the side of the triangle, use either a protractor or another source of a right angle (like the
corner of a straightedge).
c. Label the orthocenter K and estimate its coordinates from your graph.
d. Calculate the coordinates of the orthocenter algebraically. In order to get full credit, you must
show all work in an organized way on a separate sheet of paper.

Find the equations of two of the altitudes. Please write the final equation of each line on the
correct altitude on the graph. To calculate the equations of these lines, you will need to
remember how to find the slope of a line if you know the slope of a line that is perpendicular
to it.

Calculate the point of intersection of the two lines. Your solution to this system of linear
equations will be the coordinates of the orthocenter. Compare your calculated orthocenter
with your estimated orthocenter. They should be reasonably close. Otherwise, you will need
to double check your work and try again.

Your calculations may involve fractions/decimals. Use the exact coordinates, if possible.
If you must round, round your final coordinates to the nearest hundredth.
Name __________________________ Period ___________
Date _______
Geometry – Points of Concurrency of a Triangle 5.4 Centroid & Orthocenter
This sheet should be on top and stapled to the rest of your work which should include
your graph and all calculations. Your project should be ready to hand in when you come
to class on the due date (Monday, January 14).
Write your coordinates below.
COORDINATES OF VERTICES: A (
The Centroid (8 points)
Graph is neatly
Graph is correct
drawn, labeled,
and neat but
medians are
missing labels
drawn correctly
(3)
(4)
All work is
Most work is
included, is neat
included, is neat
and accurate and and accurate
shows complete
and shows
understanding (4) complete
understanding
(3)
,
), B (
,
), and C (
Graph is mostly
correct but not
neatly drawn (2)
Graph is mostly
correct but is
missing most
labels (1)
Some work is
missing but is
neat and
accurate and
shows some
understanding
(2)
Work is
disorganized but
shows some
understanding
(1)
TOTAL
The Orthocenter (8 points)
Graph is neatly
Graph is correct
drawn, labeled,
and neat but
altitudes are
missing labels
drawn correctly
(3)
(4)
All work is
Most work is
included, is neat
included, is neat
and accurate and and accurate
shows complete
and shows
understanding (4) complete
understanding
(3)
,
)
Graph is
incorrect (0)
Work does not
show
understanding, is
very
disorganized, or
is not included
(0)
_________
Graph is mostly
correct but not
neatly drawn (2)
Graph is mostly
correct but is
missing most
labels (1)
Graph is
incorrect (0)
Some work is
missing but is
neat and
accurate and
shows some
understanding
(2)
Work is
disorganized but
shows some
understanding
(1)
Work does not
show
understanding, is
very
disorganized, or
is not included
(0)
TOTAL
_________