Lesson 5

Splitting Triangle Sides with Dilation, Part 1

• Let’s draw segments connecting midpoints of the sides of triangles.

5.1: Notice and Wonder: Midpoints

Here’s a triangle $$ABC$$ with midpoints $$L, M$$, and $$N$$.

What do you notice? What do you wonder?

5.2: Dilation or Violation?

Here’s a triangle $$ABC$$. Points $$M$$ and $$N$$ are the midpoints of 2 sides.

1. Convince yourself triangle $$ABC$$ is a dilation of triangle $$AMN$$. What is the center of the dilation? What is the scale factor?
2. Convince your partner that triangle $$ABC$$ is a dilation of triangle $$AMN$$, with the center and scale factor you found.
3. With your partner, check the definition of dilation on your reference chart and make sure both of you could convince a skeptic that $$ABC$$ definitely fits the definition of dilation.
4. Convince your partner that segment $$BC$$ is twice as long as segment $$MN$$.
5. Prove that $$BC=2MN$$. Convince a skeptic.

5.3: A Little Bit Farther Now

Here’s a triangle $$ABC$$. $$M$$ is $$\frac23$$ of the way from $$A$$ to $$B$$. $$N$$ is $$\frac23$$ of the way from $$A$$ to $$C$$.

What can you say about segment $$MN$$, compared to segment $$BC$$? Provide a reason for each of your conjectures.

1. Dilate triangle $$DEF$$ using a scale factor of -1 and center $$F$$.
2. How does $$DF$$ compare to $$D'F'$$?
3. Are $$E$$, $$F$$, and $$E'$$ collinear? Explain or show your reasoning.

Summary

Let's examine a segment whose endpoints are the midpoints of 2 sides of the triangle. If $$D$$ is the midpoint of segment $$BC$$ and $$E$$ is the midpoint of segment $$BA$$, then what can we say about $$ED$$ and triangle $$ABC$$?

Segment $$ED$$ is parallel to the third side of the triangle and half the length of the third side of the triangle. For example, if $$AC=10$$, then $$ED=5$$. This happens because the entire triangle $$EBD$$ is a dilation of triangle $$ABC$$ with a scale factor of $$\frac12$$.

In triangle $$ABC$$, segment $$FG$$ divides segments $$AB$$ and $$CB$$ proportionally. In other words, $$\frac{BG}{GA}$$=$$\frac{BF}{FC}$$. Again, there is a dilation that takes triangle $$ABC$$ to triangle $$GBF$$, so $$FG$$ is parallel to $$AC$$ and we can calculate its length using the same scale factor.

Glossary Entries

• dilation

A dilation with center $$P$$ and positive scale factor $$k$$ takes a point $$A$$ along the ray $$PA$$ to another point whose distance is $$k$$ times farther away from $$P$$ than $$A$$ is.

Triangle $$A'B'C'$$ is the result of applying a dilation with center $$P$$ and scale factor 3 to triangle $$ABC$$.

• scale factor

The factor by which every length in an original figure is increased or decreased when you make a scaled copy. For example, if you draw a copy of a figure in which every length is magnified by 2, then you have a scaled copy with a scale factor of 2.