The goal of this lesson is to use segment partitioning logic to prove that the medians of a triangle intersect in a single point.
Students begin by finding midpoints of the sides of triangles, and they learn that medians are segments that connect a triangle vertex to the midpoint of the opposite side. Next, they find points that partition the medians of a triangle in a \(2:1\) ratio, observing in the process that the 3 medians of the triangle intersect in a single point. Finally, students work in groups to construct a viable argument (MP3) that proves the medians of any triangle intersect at that \(2:1\) partition point. While the intersection point of the medians isn’t given the formal name centroid in this lesson, you may choose to use that term if it is helpful.
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 16.1 Warmup: Triangle Midpoints (5 minutes)

16.2 Activity: Triangle Medians (15 minutes)
 Includes "Are you Ready for More?" extension problem
 16.3 Activity: Any Triangle’s Medians (15 minutes)
 Lesson Synthesis
 16.4 Cooldown: Intersection Point (5 minutes)
Learning goals:
 Calculate the intersection point of a triangle’s medians.
 Prove that the medians of a triangle meet at a point.
Learning goals (student facing):
 Let’s partition special line segments in triangles.
Learning targets (student facing):
 I can determine the point where the medians of a triangle intersect.
Required materials:
 Index cards
 Rulers
 Scissors
Required preparation:
 Students will use rulers as a straightedge and for taking measurements in both the Triangle Medians activity and the lesson synthesis.
 The index cards are for use in the lesson synthesis.
Glossary:
 median (geometry)  A line from a vertex of a triangle to the midpoint of the opposite side. Each dashed line in the image is a median.
 Access the complete Geometry Course Glossary.
Standards:
 This lesson builds on the standards:CCSS.HSGCO.C.10MS.GCO.10MO.G.CO.C.9CCSS.HSGGPE.B.6MS.GGPE.6MO.G.GPE.B.5
 This lesson builds towards the standards:CCSS.HSGCO.C.10MS.GCO.10MO.G.CO.C.9
IM Algebra 1, Geometry, Algebra 2 is copyright 2019 Illustrative Mathematics and licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).
The Illustrative Mathematics name and logo are not subject to the Creative Commons license and may not be used without the prior and express written consent of Illustrative Mathematics.