# Lesson 16

Weighted Averages in a Triangle

- Let’s partition special line segments in triangles.

### 16.1: Triangle Midpoints

Triangle \(ABC\) is graphed.

Find the midpoint of each side of this triangle.

### 16.2: Triangle Medians

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.

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

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

### 16.3: Any Triangle’s Medians

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.

### Summary

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 \((0,0), (0,w)\), 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.

### 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 \(\frac{1}{p}\).