Kuzas, Richmond, Tocchi – Team 6 – Centroid
Finding the Center of Mass of a Triangle
Lesson Summary:
Students will investigate how to find the center of mass by using medians of a triangle. The first
activity will have students find the center of mass using pencil and straight edge. In the second
part students will find the center of mass by using Cabri Geometry.
Key Words:
median, center of mass, midpoint, centroid
Background knowledge:
Students need to be familiar with the basic tools of Geometry software. They must also
understand midpoint and medians of a triangle.
Lesson Objectives:
1. To use the medians of a triangle to construct the center of mass
2. To understand center of mass and its basic characteristics
Materials:
Cardboard cut into the shape of a triangle, scissors, ruler, pen or pencil, string
Suggested Procedure:
Divide the class into two large groups. One group can work on the first activity (pencil and
paper) and the other group can work on the software activity. Before sending the students to the
separate activities, hold up a cardboard triangle and ask students to find the balancing point.
Students can brainstorm then try balancing the triangle on the tip of a pencil. After some
discussion, have students complete the following worksheets.
Assessments:
Collect the worksheets and evaluate the students’ responses.
Kuzas, Richmond, Tocchi – Team 6 – Centroid
Activity I. Finding the Center of Mass of a Triangle
Activity goals: To investigate how to find the center of mass, or balancing point, of a triangle.
You will need a cardboard triangle, scissors, ruler, and string.
1. Use a ruler to mark the midpoint of all three sides of the triangle and label them X, Y, and
Z.
2. Take the ruler and draw a segment from each midpoint to the opposite vertex.
3. Cut a tiny hold at the intersection of all three segments and tie a string though the hole.
Then tie a knot at the end of the string so that it does not pull back through the hole.
4. Lift the triangle by the string. What do you notice about the balance of the triangle?
The triangle is perfectly balanced.
Kuzas, Richmond, Tocchi – Team 6 – Centroid
Activity II. Finding the Center of Mass of a Triangle
Lab Goals:
1. To use the medians of a triangle to construct the center of mass using Geometry software.
2. To understand center of mass and its basic properties.
Activity:
1. Construct triangle ABC
2. Find the midpoint X, Y, and Z of sides AC , AB , and BC
3. Construct the medians, that is, the segments AZ , BY , and CX .
(use triangle tool)
(use midpoint tool)
(use segment tool)
4. Drag a vertex, A, B, or C. Is there ever a time when the medians AZ , BY , and CX do
not intersect?
No, the lines are always intersecting each other.
5. Do the midpoints X, Y, and Z remain midpoints when the vertices of the triangle are
dragged?
Yes.
6. What do you notice about point M when dragging any of the three vertices?
It moves with the vertices.
Definition: The point M where the medians of a triangle meet is called the center of mass
or centroid of the triangle.
7. Create segments from BM , ZM , CM , YM , XM , and AM .
(use segment tool)
8. Measure each new segment.
(use measure tool)
9. Calculate BM/BY, CM/CX, and AM/AZ.
(use calculate tool)
What are the results?
BM/BY = 0.67
CM/CX = 0.67
AM/AZ = 0.67
10. What do you notice about these results?
The ratio from the vertex to the midpoint and the vertex to the median point of the
opposite side for each of the lines is exactly the same.
Kuzas, Richmond, Tocchi – Team 6 – Centroid
11. From your observations thus far, what properties characterize the center of mass of a
triangle?
The three medians always meet at a single point. The point is the center of gravity of the
triangle. The medians divide the triangle into six smaller triangles.
12. Extension. Consider all possible triangles defined by any three labeled points in ABC ,
like AXM or ABM . Calculate the areas of these triangles using the area tool. What
property do you notice?
All smaller triangles have the same area as each other regardless of the shape of the
triangle. The three larger triangles also have the same area. The three larger triangles each
have an area 1/3 of ABC and the six small triangles all have an area of 1/6 of ABC .
13. If you were building a lamp with a triangular frame, where will you place the hanging
point of the lamp inside the frame so that the lamp will remain level when you hang it
up? Justify your answer.
I would place the hanging point where the three medians of the triangle meet. This point
is the center of gravity of the circle. This point always keeps the lamp level.
Kuzas, Richmond, Tocchi – Team 6 – Centroid
Journal Activity
Centroid of a Triangle
1. List all definitions and properties that you have learned in this activity.
median – A line created using the midpoint of a side of a triangle to the vertex opposite it.
center of mass (centroid) – The point created by the intersection of three median lines of
a triangle
2. Can you think of any application of the centroid?
Certainly whenever trying to build something triangular or hang something triangular, I
would use this property to figure out where the center of mass is.
3. Can you relate this topic/concept with other(s) previously study? Explain your answer.
I see relations with the Incenter of a triangle. While this center point does something
completely different than that center point, I can see similarities.