Unit 2, Lesson 12
Similarity
Accelerated 7
Preparation
Lesson Narrative
The purpose of this lesson is to define that two figures are similar if there is a sequence of translations, rotations, reflections, and dilations that takes one figure to the other. It is important to note that there are many different sequences that could show two figures are similar.
Students begin the lesson by studying pairs of triangles where some pairs are scaled copies of each other and some pairs are not. Next, in order to show that scaled copies are similar, students find and describe a sequence of transformations that takes one figure to the other using precise mathematical language (MP6).
Students also look at quadrilaterals and ways to decide whether or not they are similar. Since the transformations we have studied (translations, rotations, reflections, dilations) do not change angle measures, similar polygons will have congruent corresponding angles. Also, since dilations change all side lengths by the same scale factor, similar polygons will have proportional corresponding side lengths. While these two key ideas can determine whether two polygons are not similar, using them to determine that polygons are similar will not be formally proven until high school. In this course, similar figures are identified using transformations.
In the optional activity, students are given a card with a figure and must find another student in the room with a similar (but not congruent) figure, again explaining why their two figures are similar (MP3).
- Critique (orally) arguments that claim two polygons are similar.
- Justify (orally) the similarity of two figures using a sequence of transformations that takes one figure to the other.
Let's explore similar figures.
Required Materials
Activity 2
Required Preparation
Provide access to geometry toolkits. For the digital version of the activity, acquire devices that can run the applet.
Two figures are similar if one can fit exactly over the other after transformations.
This figure shows triangle is similar to triangle .
- Rotate triangle around point .
- Then dilate it with center point .
- The image will fit exactly over triangle .
A sequence of transformations. Triangle ABC is rotated around point B and and then dilated with center point O to fit exactly over Triangle DEF. Triangle ABC is similar to triangle DEF.