Lesson 16

In this activity, students find the midpoint of each side of a triangle. This allows for practice using an average to find the midpoint and builds the basis for the diagram in the next activity. Monitor for students who use different strategies to find the midpoint, such as counting, averaging \(x\)-coordinates then \(y\)-coordinates as in the traditional midpoint formula, and using the notation \(\frac12 A + \frac12 B\).

Invite previously selected students to share their strategies.

If not mentioned, ask students to calculate \(\frac12A + \frac12B\) and compare it to their diagram.

Students examine the medians of a particular triangle. After noticing the 3 segments appear to coincide at a point partitioning the median in a \(2:1\) ratio, students prove that is the case for this particular triangle. They will generalize this idea in the next activity.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Tell students that a median of a triangle is a line segment connecting a vertex to the midpoint of the opposite side. Ask students how many medians can be drawn in any triangle (3).

Now ask students to draw the 3 medians of the triangle in the warm-up by using a straightedge to connect the midpoint of each side to the opposite vertex. What do you notice? (The medians seem to intersect at one point.) Instruct students to label the diagram as shown here.

Invite a student to share their strategy for one point. Then ask:

• “What do you notice?” (All 3 points are the same!)
• “Have we proven that this will work for all triangles?” (No.)

Tell students they will generalize the process in the next activity.

Students generalize their observations and explanation from the previous activity to prove the medians of a triangle always coincide. It’s okay if groups don’t include all necessary details in the proof. Anything likely to be missed will be discussed in the activity synthesis.

Arrange students in groups of 3–4. Give students 2 minutes of work time to read and interpret the beginning of the problem, monitoring for groups that draw a diagram. Then, pull the class back together briefly, inviting a group to share its diagram.

A diagram is a helpful tool.

Suggest that students share their results with their groups after each step.

Make sure students understand that the \(2:1\) ratio applies to the segment that begins with the vertex and ends with the midpoint of the opposite side of the triangle.

If students struggle to find the point that partitions the median in a \(2:1\) ratio, remind them of the notation they used in earlier activities: \(\frac13A + \frac23B\). How does that apply to this problem?

The goal of the discussion is to ensure students have incorporated all the necessary elements in their proof. Here are some questions for discussion:

• “What can we conclude if we know that a single point is on 3 different lines?” (It means either the lines coincide, or the lines intersect at that point.)
• “How do we know the medians don’t coincide?” (The midpoints are all different because the sides of the triangles don’t coincide.)
• “Why did we test a \(2:1\) ratio?” (That’s the ratio we measured for the triangle in the warm-up, so we thought it might apply to all triangles.)
• “Does this proof apply to a triangle that isn’t positioned like the one in the diagram?” (Yes. We can use a sequence of rigid motions to take any triangle to an image in which a vertex is at \((0,0)\) and one side coincides with the \(x\)-axis.)