# Lesson 15

Weighted Averages

• Let’s split segments using averages and ratios.

### 15.1: Part Way: Points

For the questions in this activity, use the coordinate grid if it is helpful to you.

1. What is the midpoint of the segment connecting $$(1,2)$$ and $$(5,2)$$?
2. What is the midpoint of the segment connecting $$(5,2)$$ and $$(5,10)$$?
3. What is the midpoint of the segment connecting $$(1,2)$$ and $$(5,10)$$?

### 15.2: Part Way: Segment

Point $$A$$ has coordinates $$(2,4)$$. Point $$B$$ has coordinates $$(8,1)$$.

1. Find the point that partitions segment $$AB$$ in a $$2:1$$ ratio.
2. Calculate $$C=\frac 13 A + \frac 23 B$$.
4. For 2 new points $$K$$ and $$L$$, write an expression for the point that partitions segment $$KL$$ in a $$3:1$$ ratio.

Consider the general quadrilateral $$QRST$$ with $$Q=(0,0),R=(a,b),S=(c,d),$$ and $$T=(e,f)$$.

1. Find the midpoints of each side of this quadrilateral.
2. Show that if these midpoints are connected consecutively, the new quadrilateral formed is a parallelogram.

Here is quadrilateral $$ABCD$$.

1. Find the point that partitions segment $$AB$$ in a $$1:4$$ ratio. Label it $$B’$$.
2. Find the point that partitions segment $$AD$$ in a $$1:4$$ ratio. Label it $$D’$$.
3. Find the point that partitions segment $$AC$$ in a $$1:4$$ ratio. Label it $$C’$$.
4. Is $$AB’C’D’$$ a dilation of $$ABCD$$? Justify your answer.

### Summary

To find the midpoint of a line segment, we can average the coordinates of the endpoints. For example, to find the midpoint of the segment from $$A=(0,4)$$ to $$B=(6,7)$$, average the coordinates of $$A$$ and $$B$$: $$\left(\frac{0 + 6}{2}, \frac{4+7}{2}\right) = (3,5.5)$$. Another way to write what we just did is $$\frac12 (A+B)$$ or $$\frac12 A + \frac12 B$$.

Now, let’s find the point that is $$\frac23$$ of the way from $$A$$ to $$B$$. In other words, we’ll find point $$C$$ so that segments $$AC$$ and $$CB$$ are in a $$2:1$$ ratio.

In the horizontal direction, segment $$AB$$ stretches from $$x=0$$ to $$x=6$$. The distance from 0 to 6 is 6 units, so we calculate $$\frac23$$ of 6 to get 4. Point $$C$$ will be 4 horizontal units away from $$A$$, which means an $$x$$-coordinate of 4.

In the vertical direction, segment $$AB$$ stretches from $$y=4$$ to $$y=7$$. The distance from 4 to 7 is 3 units, so we can calculate $$\frac23$$ of 3 to get 2. Point $$C$$ must be 2 vertical units away from $$A$$, which means a $$y$$-coordinate of 6.

It is possible to do this all at once by saying $$C = \frac13 A + \frac23 B$$. This is called a weighted average. Instead of finding the point in the middle, we want to find a point closer to $$B$$ than to $$A$$. So we give point $$B$$ more weight—it has a coefficient of $$\frac23$$ rather than $$\frac12$$ as in the midpoint calculation. To calculate $$C = \frac13 A + \frac23 B$$, substitute and evaluate.

$$\frac13 A + \frac23 B$$

$$\frac13 (0,4) + \frac23 (6,7)$$

$$\left(0,\frac43 \right) + \left(4, \frac{14}{3} \right)$$

$$(4,6)$$

Either way, we found that the coordinates of $$C$$ are $$(4,6)$$.

### Glossary Entries

• 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 $$\frac{1}{p}$$.