The purpose of this discussion is for students to describe their strategies for partitioning medians of a particular triangle. Invite a student to share a strategy for finding the point that partitions one of the medians.

Display the image of the triangle with its medians. Give students 1 minute of quiet think time, and ask them to be prepared to share at least one thing that they notice and one thing that they wonder. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the image.

Things students may notice:

• The medians intersect at a single point.
• The ratio is the same for each median.
• The point of intersection partitions each median in a ratio.

Things students may wonder:

• Do the medians of a triangle always intersect in a single point?
• Does an intersection point always partition the medians in the same ratio?
• Is there something special about the ratio, or could it be any ratio for a different triangle?

If the medians intersecting in a single point does not come up during the conversation, ask students to discuss this idea. 

If students struggle to find the point that partitions the median in a ratio, remind them of the notation they used in earlier activities: . How does that apply to this problem?

The goal of the discussion is to ensure that 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 that either the lines coincide or the lines intersect at that point.)
• “How do we know that the medians don’t coincide?” (The midpoints are all different because the sides of the triangles don’t coincide.)
• “Why did we test a 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 and in which one side coincides with the -axis.)

Give students 1–2 minutes to revise their proofs with any additional information needed.