# Lesson 16

Weighted Averages in a Triangle

## 16.1: Triangle Midpoints (5 minutes)

### Warm-up

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 \(\frac12 A + \frac12 B\).

### Student Facing

Triangle \(ABC\) is graphed.

Find the midpoint of each side of this triangle.

### Student Response

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### Activity Synthesis

Invite previously selected students to share their strategies.

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

## 16.2: Triangle Medians (15 minutes)

### Activity

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).

### Launch

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.

*Representation: Develop Language and Symbols.*Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide materials to allow students to physically compare the parts of each median.

*Supports accessibility for: Conceptual processing*

### Student Facing

Your teacher will tell you how to draw and label the **medians** of the triangle in the warm-up.

- After the medians are drawn and labeled, measure all 6 segments inside the triangle using centimeters. What is the ratio of the 2 parts of each median?
- Find the coordinates of the point that partitions segment \(AN\) in a \(2:1\) ratio.
- Find the coordinates of the point that partitions segment \(BL\) in a \(2:1\) ratio.
- Find the coordinates of the point that partitions segment \(CM\) in a \(2:1\) ratio.

### Student Response

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### Student Facing

#### Are you ready for more?

In the image, \(AB\) is a median.

Find the length of \(AB\) in terms of \(a,b,\) and \(c\).

### Student Response

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### Activity Synthesis

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.

## 16.3: Any Triangle’s Medians (15 minutes)

### 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.

### Launch

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.

*Representing, Conversing: MLR7 Compare and Connect.*Use this routine to prepare students for the whole-class discussion about the proof that the medians of a triangle always coincide. After groups prove that the medians of triangle intersect at a single point, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their proofs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to justify why the proof applies to any triangle on the coordinate plane.

*Design Principle(s): Cultivate conversation*

*Engagement: Develop Effort and Persistence.*Provide reminders or checklists that focus on increasing the length of on-task orientation in the face of distractions. For example, provide a checklist of the responsibilities of each group member with pre-assigned sides and medians.

*Supports accessibility for: Attention; Social-emotional skills*

### Student Facing

The goal is to prove that the medians of any triangle intersect at a point. Suppose the vertices of a triangle are \((0,0), (w,0),\) and \((a,b)\).

- Each student in the group should choose 1 side of the triangle. If your group has 4 people, 2 can work together. Write an expression for the midpoint of the side you chose.
- Each student in the group should choose a median. Write an expression for the point that partitions each median in a \(2:1\) ratio from the vertex to the midpoint of the opposite side.
- Compare the coordinates of the point you found to those of your groupmates. What do you notice?
- Explain how these steps prove that the 3 medians of any triangle intersect at a single point.

### Student Response

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### Anticipated Misconceptions

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?

### Activity Synthesis

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.)

## Lesson Synthesis

### Lesson Synthesis

Invite students to draw a triangle on an index card and locate the point where the **medians** all intersect. Students may use folding or measuring to find midpoints. Is it possible to find the intersection point without drawing all 3 medians? (Yes. Draw 2 and find the intersection point or draw 1 and find the point which partitions the segment in a \(2:1\) ratio.)

It turns out this point is the balancing point of a triangle. Challenge students to cut out their triangles and balance them on the eraser of a pencil. If they were really precise, the triangles will balance on the tip of the pencil as well. Collecting all of these triangles and making a mobile is a fun way to decorate the classroom.

## 16.4: Cool-down - Intersection Point (5 minutes)

### Cool-Down

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## Student Lesson Summary

### Student Facing

Here is a triangle with its **medians** drawn in. A median is a line segment drawn from a vertex of a triangle to the midpoint of the opposite side. Triangles have 3 medians, with 1 for each vertex.

Notice that the medians intersect at 1 point. This point is always \(\frac23\) of the distance from a vertex to the opposite midpoint. Another way to say this is that the point of intersection, \(V\), partitions segments \(AJ,BL,\) and \(CK\) so that the ratios \(AV:VJ,BV:VL,\) and \(CV:VK\) are all \(2:1\).

We can prove this by working with a general triangle that can represent any triangle. Since any triangle can be transformed so that 1 vertex is on the origin and 1 side lies on the \(x\)-axis, we can say that our general triangle has vertices \((0,0), (w,0)\), and \((a,b)\). Through careful calculation, we can show that all 3 medians go through the point \(\left(\frac{a+w}{3},\frac{b}{3}\right)\). Therefore, the medians intersect at this point, which partitions each median in a \(2:1\) ratio from the vertex to the opposite side’s midpoint.