In the previous unit, students saw that two figures are congruent when there is a sequence of translations, rotations, and reflections that takes one figure to another. Now dilations, studied in previous lessons, are added to the possible set of “moves” taking one shape to another. Two figures are similar if there is a sequence of translations, rotations, reflections, and dilations that takes one figure to the other. When two figures are similar, there are always many different sequences that show that they are similar. One method is to apply a dilation to one figure so that the corresponding figures are congruent. Then a sequence of rigid motions will finish taking one shape to the other. Alternatively, we could translate one pair of corresponding vertices together, apply rotations and reflections to adjust the orientations, and then conclude with a dilation so that they match.
In future lessons, students will learn shortcuts for some polygons (including all triangles), but in this lesson they focus on the definition of similarity in terms of transformations. They will see that two dilations with the same scale factor but different centers differ by a translation. They will also study how transformations from polygon A to polygon B can be reversed to take polygon B to polygon A.
- Comprehend that the phrase “similar figures” (in written and spoken language) means there is a sequence of translations, rotations, reflections, and dilations that takes one figure to the other.
- Justify (orally) the similarity of two figures using a sequence of transformations that takes one figure to the other.
Let’s explore similar figures.
If you decide to do the optional “Methods for Translations and Dilations” activity, print and cut out 1 set of cards for every 2 students.
- I can apply a sequence of transformations to one figure to get a similar figure.
- I can use a sequence of transformations to explain why two figures are 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.
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