Lesson 11

Splitting Triangle Sides with Dilation, Part 2

• Let’s investigate parallel segments in triangles.

11.1: Notice and Wonder: Parallel Segments

What do you notice? What do you wonder?

11.2: Prove It: Parallel Segments

Does a line parallel to one side of a triangle always create similar triangles?

1. Create several examples. Decide if the conjecture is true or false. If it’s false, make a more specific true conjecture.
2. Find any additional information you can be sure is true.
Label it on the diagram.

3. Write an argument that would convince a skeptic that your conjecture is true.

11.3: Preponderance of Proportional Relationships

Find the length of each unlabelled side.

1. Segments $$AB$$ and $$EF$$ are parallel.
2. Segments $$BD$$ and $$FG$$ are parallel. Segment $$EG$$ is 12 units long. Segment $$EB$$ is 2.5 units long.

Find the lengths of sides $$CE,CB$$, and $$CA$$ in terms of $$x, y,$$ and $$z$$. Explain or show your reasoning.

Summary

In triangle $$ABC$$, segment $$FG$$ is parallel to segment $$AC$$. We can show that corresponding angles in triangle $$ACB$$ and triangle $$GFB$$ are congruent, so the triangles are similar by the Angle-Angle Triangle Similarity Theorem. There must be a dilation that sends triangle $$GFB$$ to triangle $$ACB$$, and so pairs of corresponding side lengths are in the same proportion. Then we can show that segment $$GF$$ divides segments $$AB$$ and $$CB$$ proportionally. In other words, $$\frac{BG}{GA}$$=$$\frac{BF}{FC}$$.

For example, suppose $$G$$ is $$\frac23$$ of the way from $$A$$ to $$B$$ and $$F$$ is $$\frac23$$ of the way from $$C$$ to $$B$$. Then if $$BA=9$$ and $$BC=12$$, we know that $$GA=6$$ and $$FC=8$$. What will $$BG$$ and $$BF$$ equal? Since $$BG=3$$ and $$BF=4$$, we know that $$\frac36=\frac48$$ and can show that $$\frac{BG}{GA}$$=$$\frac{BF}{FC}$$.

This argument holds in general. A segment in a triangle that is parallel to one side of the triangle divides the other 2 sides of the triangle proportionally.

Glossary Entries

• similar

One figure is similar to another if there is a sequence of rigid motions and dilations that takes the first figure onto the second.

Triangle $$A'B'C'$$ is similar to triangle $$ABC$$ because a rotation with center $$B$$ followed by a dilation with center $$P$$ takes $$ABC$$ to $$A'B'C'$$.