# Lesson 16

Weighted Averages in a Triangle

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

• Calculate the intersection point of a triangle’s medians.
• Prove that the medians of a triangle meet at a point.

### Student Facing

• Let’s partition special line segments in triangles.

### 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.

### Student Facing

• I can determine the point where the medians of a triangle intersect.

Building On

Building Towards

### Glossary Entries

• 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.

• opposite

Two numbers are opposites of each other if they are the same distance from 0 on the number line, but on opposite sides.

The opposite of 3 is -3 and the opposite of -5 is 5.

• point-slope form

The form of an equation for a line with slope $$m$$ through the point $$(h,k)$$. Point-slope form is usually written as $$y-k = m(x-h)$$. It can also be written as $$y = k + m(x-h)$$.

• reciprocal

If $$p$$ is a rational number that is not zero, then the reciprocal of $$p$$ is the number $$\frac{1}{p}$$.