# Lesson 9

Side Length Quotients in Similar Triangles

### Lesson Narrative

In prior lessons, students learned that similar triangles are the images of each other under a sequence of rigid transformations and dilations, and that as a result, there is a scale factor that we can use to multiply all of the side lengths in one triangle to find the corresponding side lengths in a similar triangle. In this lesson, they will discover that if you determine the quotient of a pair of side lengths in one triangle, it will be equal to the quotient of the corresponding side lengths in a similar triangle. While this fact is not limited to triangles, this lesson focuses on the particular case of triangles so that students are ready to investigate the concept of slope in upcoming lessons.

### Learning Goals

Teacher Facing

• Calculate unknown side lengths in similar triangles using the ratios of side lengths within the triangles and the scale factor between similar triangles.
• Generalize (orally) that the quotients of pairs of side lengths in similar triangles are equal.

### Student Facing

Let’s find missing side lengths in triangles.

### Student Facing

• I can decide if two triangles are similar by looking at quotients of lengths of corresponding sides.
• I can find missing side lengths in a pair of similar triangles using quotients of side lengths.

Building On

### Glossary Entries

• similar

Two figures are similar if one can fit exactly over the other after rigid transformations and dilations.

In this figure, triangle $$ABC$$ is similar to triangle $$DEF$$.

If $$ABC$$ is rotated around point $$B$$ and then dilated with center point $$O$$, then it will fit exactly over $$DEF$$. This means that they are similar.