In middle school, students learned the definition of similar figures as a pair of figures in which there is a sequence of rigid motions and dilations that takes the first figure onto the second. They used grids to confirm that their specific sequence of rigid motions would work for the two given figures. In a previous unit, students used the definitions of rigid transformations to reason about why any triangles with all corresponding parts congruent could be taken onto one another using rigid motions, and they proved triangle congruence theorems using rigid transformations to show that any triangles with the given criteria could be taken onto each other with the same sequence of rigid motions.
In this lesson, students develop an informal understanding of how to use rigid transformations and dilations to show the similarity of any pair of triangles with all pairs of corresponding angles congruent and all pairs of corresponding side lengths proportional. Students look at specific examples and try to come up with a generalized method that works for any pair of triangles that meets the criteria, much like they did with rigid transformations in the previous unit (MP8). They draw on their understanding of the relationship between congruence and rigid transformations and extend that to similarity and dilations (MP7).
- Comprehend the definition of similarity in terms of dilations and rigid transformations.
- Generate similarity statements about similar figures and identify proportional relationships among corresponding side lengths (in writing).
- Let’s identify similar figures.
- I can write similarity statements.
- I know the definition of similarity.
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 \(A'B'C'\).